7.7 Coupled-Cluster Excited-State and Open-Shell Methods

EOM-CC and most of the CI codes are part of CCMAN and CCMAN2. CCMAN is a legacy code which is being phased out. All new developments and performance-enhancing features are implemented in CCMAN2. Some options behave differently in the two modules. Below we make an effort to mark which features are available in legacy code only.

7.7.1 Excited States via EOM-EE-CCSD

One can describe electronically excited states at a level of theory similar to that associated with coupled-cluster theory for the ground state by applying either linear response theory[Koch and Jørgensen(1990)] or equation-of-motion methods.[Stanton and Bartlett(1993)] A number of groups have demonstrated that excitation energies based on a coupled-cluster singles and doubles ground state are generally very accurate for states that are primarily single electron promotions. The error observed in calculated excitation energies to such states is typically 0.1–0.2 eV, with 0.3 eV as a conservative estimate, including both valence and Rydberg excited states. This, of course, assumes that a basis set large and flexible enough to describe the valence and Rydberg states is employed. The accuracy of excited state coupled-cluster methods is much lower for excited states that involve a substantial double excitation character, where errors may be 1 eV or even more. Such errors arise because the description of electron correlation of an excited state with substantial double excitation character requires higher truncation of the excitation operator. The description of these states can be improved by including triple excitations, as in EOM(2,3).

Q-Chem includes coupled-cluster methods for excited states based on the coupled cluster singles and doubles (CCSD) method described earlier. CCMAN also includes the optimized orbital coupled-cluster doubles (OD) variant. OD excitation energies have been shown to be essentially identical in numerical performance to CCSD excited states.[Krylov et al.(2000)Krylov, Sherrill, and Head-Gordon]

These methods, while far more computationally expensive than TDDFT, are nevertheless useful as proven high accuracy methods for the study of excited states of small molecules. Moreover, they are capable of describing both valence and Rydberg excited states, as well as states of a charge-transfer character. Also, when studying a series of related molecules it can be very useful to compare the performance of TDDFT and coupled-cluster theory for at least a small example to understand its performance. Along similar lines, the CIS(D) method described earlier as an economical correlation energy correction to CIS excitation energies is in fact an approximation to EOM-CCSD. It is useful to assess the performance of CIS(D) for a class of problems by benchmarking against the full coupled-cluster treatment. Finally, Q-Chem includes extensions of EOM methods to treat ionized or electron attachment systems, as well as di- and tri-radicals.

EOM-EE $\Psi (M_ S=0)=R(M_ S=0) \Psi _0(M_ S=0)$

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EOM-IP $\Psi (N)=R(-1) \Psi _0(N+1)$

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EOM-EA $\Psi (N)=R(+1) \Psi _0(N-1)$

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EOM-SF $\Psi (M_ S=0)=R(M_ S=-1) \Psi _0(M_ S=1)$

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Figure 7.1: In the EOM formalism, target states $\Psi$ are described as excitations from a reference state $\Psi _0$ : $\Psi =R \Psi _0$ , where $R$ is a general excitation operator. Different EOM models are defined by choosing the reference and the form of the operator $R$ . In the EOM models for electronically excited states (EOM-EE, upper panel), the reference is the closed-shell ground state Hartree-Fock determinant, and the operator $R$ conserves the number of $\alpha$ and $\beta$ electrons. Note that two-configurational open-shell singlets can be correctly described by EOM-EE since both leading determinants appear as single electron excitations. The second and third panels present the EOM-IP/EA models. The reference states for EOM-IP/EA are determinants for $N+1$ / $N-1$ electron states, and the excitation operator $R$ is ionizing or electron-attaching, respectively. Note that both the EOM-IP and EOM-EA sets of determinants are spin-complete and balanced with respect to the target multi-configurational ground and excited states of doublet radicals. Finally, the EOM-SF method (the lowest panel) employs the high-spin triplet state as a reference, and the operator $R$ includes spin-flip, i.e., does not conserve the number of $\alpha$ and $\beta$ electrons. All the determinants present in the target low-spin states appear as single excitations, which ensures their balanced treatment both in the limit of large and small HOMO/LUMO gaps. Other EOM methods available in Q-Chem are EOM-2SF and EOM-DIP.

7.7.2 EOM-XX-CCSD and CI Suite of Methods

Q-Chem features the most complete set of EOM-CCSD models,[Krylov(2008)] enabling accurate, robust, and efficient calculations of electronically excited states (EOM-EE-CCSD or EOM-EE-OD);[Sekino and Bartlett(1984), Koch et al.(1990)Koch, Jensen, Jørgensen, and Helgaker, Stanton and Bartlett(1993), Krylov et al.(2000)Krylov, Sherrill, and Head-Gordon, Levchenko and Krylov(2004)]; ground and excited states of diradicals and triradicals (EOM-SF-CCSD and EOM-SF-OD);[Krylov(2001), Levchenko and Krylov(2004)] ionization potentials and electron attachment energies, as well as problematic doublet radicals and cation or anion radicals (EOM-IP/EA-CCSD).[Sinha et al.(1989)Sinha, Mukhopadhya, Chaudhuri, and Mukherjee, Stanton and Gauss(1994), Nooijen and Bartlett(1995)] The EOM-DIP-CCSD and EOM-2SF-CCSD methods are available as well. Conceptually, EOM is very similar to configuration interaction (CI): target EOM states are found by diagonalizing the similarity transformed Hamiltonian $\bar H= e^{-T}H e^{T}$ ,

$\begin{equation} \bar H R = ER, \end{equation}$

(7.47)

where $T$ and $R$ are general excitation operators with respect to the reference determinant $|\Phi _0\ensuremath{\rangle }$ . In the EOM-CCSD models, $T$ and $R$ are truncated at single and double excitations, and the amplitudes $T$ satisfy the CC equations for the reference state $|\Phi _0\ensuremath{\rangle }$ :

	$\displaystyle \ensuremath{\langle }\Phi _ i^ a\|\bar H \| \Phi _0\ensuremath{\rangle }$	$\displaystyle =$	$\displaystyle 0$		(7.48)
	$\displaystyle \ensuremath{\langle }\Phi _{ij}^{ab}\|\bar H \| \Phi _0\ensuremath{\rangle }$	$\displaystyle =$	$\displaystyle 0$		(7.49)

The computational scaling of EOM-CCSD and CISD methods is identical, i.e., ${\cal {O}}({N^6})$ , however EOM-CCSD is numerically superior to CISD because correlation effects are “folded in” in the transformed Hamiltonian, and because EOM-CCSD is rigorously size-intensive.

By combining different types of excitation operators and references $|\Phi _0\ensuremath{\rangle }$ , different groups of target states can be accessed as explained in Fig. 7.1. For example, electronically excited states can be described when the reference $|\Phi _0\ensuremath{\rangle }$ corresponds to the ground state wave function, and operators $R$ conserve the number of electrons and a total spin.[Stanton and Bartlett(1993)] In the ionized/electron attached EOM models,[Stanton and Gauss(1994), Nooijen and Bartlett(1995)] operators $R$ are not electron conserving (i.e., include different number of creation and annihilation operators)—these models can accurately treat ground and excited states of doublet radicals and some other open-shell systems. For example, singly ionized EOM methods, i.e., EOM-IP-CCSD and EOM-EA-CCSD, have proven very useful for doublet radicals whose theoretical treatment is often plagued by symmetry breaking. Finally, the EOM-SF method[Krylov(2001), Levchenko and Krylov(2004)] in which the excitation operators include spin-flip allows one to access diradicals, triradicals, and bond-breaking.[Krylov(2017)]

Q-Chem features EOM-EE/SF/IP/EA/DIP/DSF-CCSD methods for both closed and open-shell references (RHF/UHF/ROHF), including frozen core/virtual options. For EE, SF, IP, and EA, a more economical flavor of EOM-CCSD is available (EOM-MP2 family of methods). All EOM models take full advantage of molecular point group symmetry. Analytic gradients are available for RHF and UHF references, for the full orbital space, and with frozen core/virtual orbitals.[Levchenko et al.(2005)Levchenko, Wang, and Krylov] Properties calculations (permanent and transition dipole moments, $\ensuremath{\langle }S^2\ensuremath{\rangle }$ , $\ensuremath{\langle }R^2 \ensuremath{\rangle }$ , etc.) are also available. The current implementation of the EOM-XX-CCSD methods enables calculations of medium-size molecules, e.g., up to 15–20 heavy atoms. Using RI approximation 6.8.5 or Cholesky decomposition 6.8.6 helps to reduce integral transformation time and disk usage enabling calculations on much larger systems. EOM-MP2 and EOM-MP2t variants are also less computationally demanding. The computational cost of EOM-IP calculations can be considerably reduced (with negligible decline in accuracy) by truncating virtual orbital space using FNO scheme (see Section 7.7.8).

Legacy features available in CCMAN. The CCMAN module of Q-Chem includes two implementations of EOM-IP-CCSD. The proper implementation[Pieniazek et al.(2008)Pieniazek, Bradforth, and Krylov] is used by default is more efficient and robust. The EOM_FAKE_IPEA keyword invokes is a pilot implementation in which EOM-IP-CCSD calculation is set up by adding a very diffuse orbital to a requested basis set, and by solving EOM-EE-CCSD equations for the target states that include excitations of an electron to this diffuse orbital. The implementation of EOM-EA-CCSD in CCMAN also uses this trick. Fake IP/EA calculations are only recommended for Dyson orbital calculations and debug purposes. (CCMAN2 features proper implementations of EOM-IP and EOM-EA (including Dyson orbitals)).

A more economical CI variant of EOM-IP-CCSD, IP-CISD is also available in CCMAN. This is an N $^5$ approximation of IP-CCSD, and can be used for geometry optimizations of problematic doublet states.[Golubeva et al.(2009)Golubeva, Pieniazek, and Krylov]

7.7.3 Spin-Flip Methods for Di- and Triradicals

The spin-flip method[Krylov(2001), Krylov(2002), Krylov(2006)] addresses the bond-breaking problem associated with a single-determinant description of the wave function. Both closed and open shell singlet states are described within a single reference as spin-flipping, (e.g., $\alpha \rightarrow \beta$ excitations from the triplet reference state, for which both dynamical and non-dynamical correlation effects are smaller than for the corresponding singlet state. This is because the exchange hole, which arises from the Pauli exclusion between same-spin electrons, partially compensates for the poor description of the coulomb hole by the mean-field Hartree-Fock model. Furthermore, because two $\alpha$ electrons cannot form a bond, no bond breaking occurs as the internuclear distance is stretched, and the triplet wave function remains essentially single-reference in character. The spin-flip approach has also proved useful in the description of di- and tri-radicals as well as some problematic doublet states.

The spin-flip method is available for the CIS, CIS(D), CISD, CISDT, OD, CCSD, and EOM-(2,3) levels of theory and the spin complete SF-XCIS (see Section 7.2.4). An N $^7$ non-iterative triples corrections are also available. For the OD and CCSD models, the following non-relaxed properties are also available: dipoles, transition dipoles, eigenvalues of the spin-squared operator ( $\langle S^2\rangle$ ), and densities. Analytic gradients are also for SF-CIS and EOM-SF-CCSD methods. To invoke a spin-flip calculation the SF_STATES $rem should be used, along with the associated $rem settings for the chosen level of correlation by using METHOD (recommended) or using older keywords (CORRELATION, and, optionally, EOM_CORR). Note that the high multiplicity triplet or quartet reference states should be used.

Several double SF methods have also been implemented.[Casanova et al.(2009)Casanova, Slipchenko, Krylov, and Head-Gordon] To invoke these methods, use DSF_STATES.

7.7.4 EOM-DIP-CCSD

Double-ionization potential (DIP) is another non-electron-conserving variant of EOM-CCSD.[Wladyslawski and Nooijen(2002), Kuś and Krylov(2011), Kuś and Krylov(2012)] In DIP, target states are reached by detaching two electrons from the reference state:

$\begin{equation} \Psi _ k = R_{N-2} \Psi _0 (N+2), \protect \label{DIP:WF} \end{equation}$

(7.50)

and the excitation operator $R$ has the following form:

$\displaystyle \label{eq:R-IP} R$	$\displaystyle =$	$\displaystyle R_1 + R_2,$	(7.51)
$\displaystyle R_1$	$\displaystyle =$	$\displaystyle 1/2 \sum _{ij} r_{ij} ji,$	(7.52)
$\displaystyle R_2$	$\displaystyle =$	$\displaystyle 1/6 \sum _{ijka} r_{ijk}^{a} a^\dag kji.$	(7.53)

As a reference state in the EOM-DIP calculations one usually takes a well-behaved closed-shell state. EOM-DIP is a useful tool for describing molecules with electronic degeneracies of the type “ $2n-2$ electrons on $n$ degenerate orbitals”. The simplest examples of such systems are diradicals with two-electrons-on-two-orbitals pattern. Moreover, DIP is a preferred method for four-electrons-on-three-orbitals wave functions.

Accuracy of the EOM-DIP-CCSD method is similar to accuracy of other EOM-CCSD models, i.e., 0.1–0.3 eV. The scaling of EOM-DIP-CCSD is ${\cal {O}}({N^6})$ , analogous to that of other EOM-CCSD methods. However, its computational cost is less compared to, e.g., EOM-EE-CCSD, and it increases more slowly with the basis set size. An EOM-DIP calculation is invoked by using DIP_STATES, or DIP_SINGLETS and DIP_TRIPLETS.

7.7.5 EOM-CC Calculations of Core-Level States: Core-Valence Separation within EOM-CCSD

The core-valence separation (CVS) approximation[Cederbaum et al.(1980)Cederbaum, Domcke, and Schirmer] allows one to extend standard methods for excited and ionized states to the core-level states. In this approach, the excitations involving core electrons are decoupled from the rest of the configurational space. This allows one to reduce computational costs and decouple the highly excited core states from the continuum. Currently, CVS is implemented within EOM-EE-CCSD for energies and transition properties (oscillator strengths, NTOs, exciton descriptors). CVS-EOM-EE-CCSD can be used to model NEXAFS.

In Q-Chem, a slightly different version of CVS-EOM-EE-CCSD than the original theory by Coriani and Koch[Coriani and Koch(2015)] is implemented: the reference coupled-cluster amplitudes do not include core electrons[Vidal et al.()Vidal, Feng, Epifanovsky, Krylov, and Coriani]. To distinguish this method from the original[Coriani and Koch(2015)], below we refer to the Q-Chem implementation as frozen-core-ground-state/core-valence-separated EOM (FC-CVS-EOM) approach.[Vidal et al.()Vidal, Feng, Epifanovsky, Krylov, and Coriani]

In the FC-CVS-EOM approach the ground-state parameters (amplitudes and Lagrangian multipliers) are computed within the frozen-core approximation, whereas the core-excitation energies and strengths are obtained imposing that at least one index in the EOM excitation operators refer to a core occupied orbital.

To ensure the best convergence of EOM equations, the calculation is edge-specific with respect to the highest lying edges (or deepest lying core orbitals): the frozen-core and CVS spaces are selected for each edge such that the core orbitals we are addressing in the excited state calculations are explicitly frozen in the ground state calculation and specifically included in the EOM calculation. Examples 7.7.5.1 and 7.7.5.1 below illustrate this point.

To invoke the CVS approximation, use METHOD=CCSD and CVS_EE_STATES instead of EE_STATES to specify the desired target states (likewise, CVS_EE_SINGLETS and CVS_EE_TRIPLETS can be used in exactly the same way as in regular EOM calculations). CVS-EOM-CCSD is only available with CCMAN2.

Note: Core electrons must be frozen in CVS-EOM calculations. The exact definition of the core depends on the edge.

7.7.5.1 Examples

In example 7.7.5.1, the 1 $s$ orbital on the oxygen atom is frozen in the CCSD calculation (N_FROZEN_CORE = FC). In the EOM calculation, the CVS approximation is invoked (CVS_EE_SINGLETS), so that the core-excitation energies are obtained as the lowest excitations. The calculation of the oscillator strengths is activated by selecting CC_TRANS_PROP=1 and the libwfa analysis is invoked by STATE_ANALYSIS = TRUE (see Section 11.2.6).

Example 7.126 FC-CVS-EOM-CCSD calculation of the first six dipole allowed core excitation energies and their intensities at the oxygen edge of water. Wave-function analysis is also performed.

$molecule
0 1
O  0.0000  0.0000  0.1173
H  0.0000  0.7572 -0.4692
H  0.0000 -0.7572 -0.4692
$end

$rem
    method           = eom-ccsd
    cvs_ee_singlets  = [3,0,2,1]
    basis            = aug-cc-pVDZ
    n_frozen_core    = fc
    CC_TRANS_PROP    = true    
    eom_preconv_singles = true
    state_analysis = true !invoke libwa to compute NTOs and exciton descriptors
! libwa controls below
    molden_format = true
    nto_pairs     = 3
    pop_mulliken  = true
$end

Example 7.7.5.1 illustrates CVS-EOM-EE-CCSD calculations in a two-edge molecule (CO). In the present implementation, the calculation should be done separately for each edge. The first job computes carbon-edge states. Since the carbon $1s$ orbital is the highest in energy (among the core $1s$ orbitals of the molecule), the input for the C-edge is similar to example 7.7.5.1. Both the oxygen’s and the carbon’s $1s$ orbitals are frozen in the reference CCSD calculation. In the EOM part, the carbon core-excited states are automatically selected. In this case, using default frozen core settings (N_FROZEN_CORE = FC) is equivalent to specifying N_FROZEN_CORE = 2. In the second input, the oxygen edge is computed. As the core-orbitals of oxygen lie deeper, the frozen core and CVS selection specifically targets the oxygen edge by using a smaller core. The 1 $s$ orbital of the oxygen atom is selected by N_FROZEN_CORE=1. If the molecule has other edges, the deepest lying core orbitals, up to and including those of the edge of interest, should be selected by an appropriate value of N_FROZEN_CORE.

Example 7.127 FC-CVS-EOM-EE-CCSD calculations of the first two dipole allowed core excitation energies per irreducible representation and their intensities at the carbon and oxygen edges of carbon monoxide.

$comment
CO, carbon edge
$end

$molecule
0 1
O  0.0000  0.0000   0.913973
C  0.0000  0.0000   -1.218243
$end

$rem
    input_bohr = true
    method     = eom-ccsd
    cvs_ee_singlets = [2,0,2,2]
    basis           = aug-cc-pVDZ
    n_frozen_core   = fc
    eom_preconv_singles = true
    CC_TRANS_PROP    = true
$end

@@@
$comment
CO, oxygen edge
$end

$molecule
read
$end

$rem
    method           = eom-ccsd
    cvs_ee_singlets  = [2,0,2,2]
    basis            = aug-cc-pVDZ
    n_frozen_core    = 1
    eom_preconv_singles = true
    CC_TRANS_PROP = true
$end

7.7.6 EOM-CC Calculations of Metastable States: Super-Excited Electronic States, Temporary Anions, and More

While conventional coupled-cluster and equation-of-motion methods allow one to tackle electronic structure ranging from well-behaved closed shell molecules to various open-shell and electronically excited species,[Krylov(2008)] meta-stable electronic states, so-called resonances, present a difficult case for theory. By using complex scaling and complex absorbing potential techniques, we extended these powerful methods to describe auto-ionizing states, such as transient anions, highly excited electronic states, and core-ionized species.[Bravaya et al.(2013)Bravaya, Zuev, Epifanovsky, and Krylov, Jagau et al.(2014)Jagau, Zuev, Bravaya, Epifanovsky, and Krylov, Jagau et al.(2017)Jagau, Bravaya, and Krylov] In addition, users can employ stabilization techniques using charged sphere and scaled atomic charges options.[Kuś and Krylov(2012)] These methods are only available within CCMAN2. The complex CC/EOM code is engaged by COMPLEX_CCMAN; the specific parameters should be specified in the $complex_ccman section.

COMPLEX_CCMAN

Requests complex-scaled or CAP-augmented CC/EOM calculations.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE

Engage complex CC/EOM code.

RECOMMENDATION:

Not available in CCMAN. Need to specify CAP strength or complex-scaling parameter in $complex_ccman section.

The $complex_ccman section is used to specify the details of the complex-scaled/CAP calculations, as illustrated below. If user specifies CS_THETA, complex scaling calculation is performed.

$complex_ccman
CS_THETA  10   Complex-scaling parameter theta=0.01, r->r exp(-i*theta)
CS_ALPHA  10   Real part of the scaling parameter alpha=0.01, 
!              r->alpha r exp(-itheta)
$end

Alternatively, for CAP calculations, the CAP parameters need to be specified.

$complex_ccman
CAP_ETA 1000  CAP strength in 10-5 a.u. (0.01)
CAP_X 2760  CAP onset along X in 10^-3 bohr (2.76 bohr)
CAP_Y 2760  CAP onset along Y in 10^-3 bohr (2.76 bohr)
CAP_Z 4880  CAP onset along Z in 10^-3 bohr (4.88 bohr)
CAP_TYPE 1  Use cuboid cap (CAP_TYPE=0 will use spherical CAP) 
$end

CS_THETA is specified in radian $\times$ 10 $^{-3}$ . CS_ALPHA, CAP_X/Y/Z are specified in a.u. $\times$ 10 $^{-3}$ , i.e., CS_THETA = 10 means $\theta$ =0.01; CAP_ETA is specified in a.u. $\times$ 10 $^{-5}$ . The CAP is calculated by numerical integration, the default grid is 000099000590. For testing the accuracy of numerical integration, the numerical overlap matrix is calculated and compared to the analytical one. If the performance of the default grid is poor, the grid type can be changed using the keyword XC_GRID (see Section 5.5 for further details). When CAP calculations are performed, CC_EOM_PROP=1 by default; this is necessary for calculating first-order perturbative correction.

Advanced users may find the following options useful. Several ways of conducing complex calculations are possible, i.e., complex scaling/CAPs can be either engaged at all levels (HF, CCSD, EOM), or not. By default, if COMPLEX_CCMAN is specified, the EOM calculations are conducted using complex code. Other parameters are set up as follows:

$complex_ccman
CS_HF=true 
CS_CCSD=true 
$end

Alternatively, the user can disable complex HF. These options are experimental and should only be used by advanced users. For CAP-EOM-CC, only CS_HF = TRUE and CS_CCSD = TRUE is implemented.

Non-iterative triples corrections are available for all complex scaled and CAP-augmented CC/EOM-CC models and requested in analogy to regular CC/EOM-CC (see Section 7.7.21 for details).

Molecular properties and transition moments are requested for complex scaled or CAP-augmented CC/EOM-CC calculations in analogy to regular CC/EOM-CC (see Section 7.7.16 for details). Natural orbitals and natural transition orbitals can be computed and the exciton wave-functions can be analyzed, similarly to real-valued EOM-CCSD (same keywords are used to invoke the analysis). Analytic gradients are available for complex CC/EOM-CC only for cuboid CAPs (CAP_TYPE = 1) introduced at the HF level (CS_HF = TRUE), as described in Ref. Benda:2017. The frozen core approximation is disabled for CAP-CC/EOM-CC gradient calculations. Geometry optimization can be requested in analogy to regular CC/EOM-CC (see Section 7.7.16 for details).

7.7.7 Charge Stabilization for EOM-DIP and Other Methods

The performance of EOM-DIP deteriorates when the reference state is unstable with respect to electron-detachment,[Kuś and Krylov(2011), Kuś and Krylov(2012)] which is usually the case for dianion references employed to describe neutral diradicals by EOM-DIP. Similar problems are encountered by all excited-state methods when dealing with excited states lying above ionization or electron-detachment thresholds.

To remedy this problem, one can employ charge stabilization methods, as described in Refs. Kus:2011, Kus:2012. In this approach (which can also be used with any other electronic structure method implemented in Q-Chem), an additional Coulomb potential is introduced to stabilize unstable wave functions. The following keywords invoke stabilization potentials: SCALE_NUCLEAR_CHARGE and ADD_CHARGED_CAGE. In the former case, the potential is generated by increasing nuclear charges by a specified amount. In the latter, the potential is generated by a cage built out of point charges comprising the molecule. There are two cages available: dodecahedral and spherical. The shape, radius, number of points, and the total charge of the cage are set by the user.

Note: A perturbative correction estimating the effect of the external Coulomb potential on EOM energy will be computed when target state densities are calculated, e.g., when CC_EOM_PROP = TRUE.

Note: Charge stabilization techniques can be used with other methods such as EOM-EE, CIS, and TDDFT to improve the description of resonances. It can also be employed to describe meta-stable ground states.

7.7.8 Frozen Natural Orbitals in CC and IP-CC Calculations

Large computational savings are possible if the virtual space is truncated using the frozen natural orbital (FNO) approach (see Section 6.11). Extension of the FNO approach to ionized states within EOM-CC formalism was recently introduced and benchmarked.[Landau et al.(2010)Landau, Khistyaev, Dolgikh, and Krylov] In addition to ground-state coupled-cluster calculations, FNOs can also be used in EOM-IP-CCSD, EOM-IP-CCSD(dT/fT) and EOM-IP-CC(2,3). In IP-CC the FNOs are computed for the reference (neutral) state and then are used to describe several target (ionized) states of interest. Different truncation scheme are described in Section 6.11.

7.7.9 Approximate EOM-CC Methods: EOM-MP2 and EOM-MP2T

Approximate EOM-CCSD models with $T$ -amplitudes obtained at the MP2 level offer reduced computational cost compared to the full EOM-CCSD since the computationally demanding $\mathcal{O}(N^6)$ CCSD step is eliminated from the calculation. Two methods of this type are implemented in Q-Chem. The first is invoked with the keyword METHOD = EOM-MP2. Its formulation and implementation follow the original EOM-CCSD(2) approach developed by Stanton and coworkers.[Stanton and Gauss(1995)] The second method can be requested with the METHOD = EOM-MP2T keyword and is similar to EOM-MP2, but it accounts for the additional terms in $\bar{H}$ that appear because the MP2 $T-$ amplitudes do not satisfy the CCSD equations. EOM-MP2 ansatz is implemented for IP/EA/EE/SF energies, state properties, and interstate properties (EOM-EOM, but not REF-EOM). EOM-MP2t is available for the IP/EE/EA energy calculations only.

7.7.10 Approximate EOM-CC Methods: EOM-CCSD-S(D) and EOM-MP2-S(D)

These are very light-weight EOM methods in which the EOM problem is solved in the singles block and the effect of doubles is evaluated perturbatively. The $\bar{H}$ is evaluated by using either CCSD or MP2 amplitudes, just as in the regular EOM calculations. The EOM-MP2-S(D) method, which is similar in level of correlation treatment to SOS-CIS(D), is particularly fast. These methods are implemented for IP and EE states. For valence states, the errors for absolute ionization or excitation energies against regular EOM-CCSD are about 0.4 eV and appear to be systematically blue-shifted; the EOM-EOM energy gaps look better. The calculations are set as in regular EOM-EE/IP, but using method = EOM-CCSD-SD(D) or method = EOM-MP2-SD(D). State properties and EOM-EOM transition properties can be computed using these methods (reference-EOM properties are not yet implemented). These methods are designed for treating core-level states.[Sadybekov and Krylov(2017)]

Note: These methods are still in the experimental stage.

7.7.11 Implicit solvent models in EOM-CC/MP2 calculations.

Vertical excitation/ionization/attachment energies can be computed for all EOM-CC/MP2 methods using a non-equilibrium C-PCM model. To perform a PCM-EOM calculation, one has to invoke the PCM (SOLVENT_METHOD to PCM in the $rem block) and specify the solvent parameters, i.e. the dielectric constant $\epsilon$ and the squared refractive index $n^2$ (DIELECTRIC and DIELECTRIC_INFI in the $solvent block). If nothing is given, the parameters for water will be used by default. For EOM methods, only the simplest model, C-PCM, is implemented. More sophisticated flavors of PCM are available for ADC methods (see Section 7.8.7). For a detailed description of PCM theory, see Sections 7.8.7, 12.2.2 and 12.2.3.

Note: Only energies and unrelaxed properties can be computed (no gradient).

Note: Symmetry is turned off for C-CPM calculations.

7.7.12 EOM-CC Jobs: Controlling Guess Formation and Iterative Diagonalizers

An EOM-CC eigen-problem is solved by an iterative diagonalization procedure that avoids full diagonalization and only looks for several eigen-states, as specified by the XX_STATES keywords.

The default procedure is based on the modified Davidson diagonalization algorithm, as explained in Ref. Levchenko:2004. In addition to several keywords that control the convergence of algorithm, memory usage, and fine details of its execution, there are several important keywords that allow user to specify how the target state selection will be performed.

By default, the diagonalization looks for several lowest eigenstates, as specified by XX_STATES. The guess vectors are generated as singly excited determinants selected by using Koopmans’ theorem; the number of guess vectors is equal to the number of target states. If necessary, the user can increase the number of singly excited guess vectors (EOM_NGUESS_SINGLES) and include doubly excited guess vectors (EOM_NGUESS_DOUBLES).

Note: In CCMAN2, if there is not enough singly excited guess vectors, the algorithm adds doubly excited guess vectors. In CCMAN, doubly excited guess vectors are generated only if EOM_NGUESS_DOUBLES is invoked.

The user can request to pre-converge singles (solve the equations in singles-only block of the Hamiltonian. This is done by using EOM_PRECONV_SINGLES.

Note: In CCMAN, the user can pre-converge both singles and doubles blocks (EOM_PRECONV_SINGLES and EOM_PRECONV_DOUBLES).

If a state (or several states) of a particular character is desired (e.g., $\rm HOMO\rightarrow LUMO+10$ excitation or $\rm HOMO-10$ ionization), the user can specify this by using EOM_USER_GUESS keyword and $eom_user_guess section, as illustrated by an example below. The algorithm will attempt to find an eigenstate that has the maximum overlap with this guess vector. The multiplicity of the state is determined as in the regular calculations, by using the XX_SINGLETS and EE_TRIPLETS keywords. This option is useful for looking for high-lying states such as core-ionized or core-excited states. It is only available with CCMAN2.

The examples below illustrate how to use user-specified guess in EOM calculations:

$eom_user_guess
4  Corresponds to 4(OCC)->5(VIRT) transition. 
5
$end

$eom_user_guess
1  5   Ex. states corresponding to 1(OCC)->5(VIRT) and 1(OCC)->6(VIRT) 
1  6
$end

In IP/EA calculations, only one set of orbitals is specified:

$eom_user_guess
4 5 6
$end

If IP_STATES is specified, this will invoke calculation of the EOM-IP states corresponding to the ionization from 4th, 5th, and 6th occupied MOs. If EA_STATES is requested, then EOM-EA equations will be solved for a root corresponding to electron-attachment to the 4th, 5th, and 6th virtual MOs.

For these options to work correctly, user should make sure that XX_STATES requests a sufficient number of states. In case of symmetry, one can request several states in each irrep, but the algorithm will only compute those states which are consistent with the user guess orbitals.

Alternatively, the user can specify an energy shift by EOM_SHIFT. In this case, the solver looks for the XX_STATES eigenstates that are closest to this energy; the guess vectors are generated accordingly, using Koopmans’ theorem. This option is useful when highly excited states (i.e., interior eigenstates) are desired.

7.7.13 Equation-of-Motion Coupled-Cluster Job Control

It is important to ensure there are sufficient resources available for the necessary integral calculations and transformations. For CCMAN/CCMAN2 algorithms, these resources are controlled using the $rem variables CC_MEMORY, MEM_STATIC and MEM_TOTAL (see Section 6.14).

The exact flavor of correlation treatment within equation-of-motion methods is defined by METHOD (see Section 7.1). For EOM-CCSD, once should set METHOD to EOM-CCSD, for EOM-MP2, METHOD = EOM-CCSD, etc.. In addition, a specification of the number of target states is required through XX_STATES (XX designates the type of the target states, e.g., EE, SF, IP, EA, DIP, DSF, etc.). Users must be aware of the point group symmetry of the system being studied and also the symmetry of the initial and target states of interest, as well as symmetry of transition. It is possible to turn off the use of symmetry by CC_SYMMETRY. If set to FALSE the molecule will be treated as having $C_1$ symmetry and all states will be of $A$ symmetry.

Note: (1) In finite-difference calculations, the symmetry is turned off automatically, and the user must ensure that XX_STATES is adjusted accordingly.
(2) In CCMAN, mixing different EOM models in a single calculation is only allowed in Dyson orbitals calculations. In CCMAN2, different types of target states can be requested in a single calculation.

7.7.13.1 Alternative way to set up EOM calculations

Below we describe alternative way to specify correlation treatment in EOM-CC/CI calculations. These keywords will be eventually phased out. By default, the level of correlation of the EOM part of the wave function (i.e., maximum excitation level in the EOM operators $R$ ) is set to match CORRELATION, however, one can mix different correlation levels for the reference and EOM states by using EOM_CORR. To request a CI calculation, set CORRELATION = CI and select type of CI expansion by EOM_CORR. The table below shows default and allowed CORRELATION and EOM_CORR combinations.

CORRELATION	Default	Allowed	Target states	CCMAN /
	EOM_CORR	EOM_CORR		CCMAN2
CI	none	CIS, CIS(D)	EE, SF	y/n
		CISD	EE, SF, IP	y/n
		SDT, DT	EE, SF, DSF	y/n
CIS(D)	CIS(D)	N/A	EE, SF	y/n
CCSD, OD	CISD		EE, SF, IP, EA, DIP	y/y
		SD(fT)	EE, IP, EA	n/y
		SD(dT), SD(fT)	EE, SF, fake IP/EA	y/n
		SD(dT), SD(fT), SD(sT)	IP	y/n
		SDT, DT	EE, SF, IP, EA, DIP, DSF	y/n

Table 7.1: Default and allowed CORRELATION and EOM_CORR combinations as well as valid target state types. The last column shows if a method is available in CCMAN or CCMAN2.

Table 7.1 shows the correct combinations of CORRELATION and EOM_CORR for standard EOM and CI models.

Method	CORRELATION	EOM_CORR	Target states selection
CIS	CI	CIS	EE_STATES
			EE_SNGLETS, EE_TRIPLETS
SF-CIS	CI	CIS	SF_STATES
CIS(D)	CI	CIS(D)	EE_STATES
			EE_SNGLETS, EE_TRIPLETS
SF-CIS(D)	CI	CIS(D)	SF_STATES
CISD	CI	CISD	EE_STATES
			EE_SNGLETS, EE_TRIPLETS
SF-CISD	CI	CISD	SF_STATES
IP-CISD	CI	CISD	IP_STATES
CISDT	CI	SDT	EE_STATES
			EE_SNGLETS, EE_TRIPLETS
SF-CISDT	CI	SDT or DT	SF_STATES
EOM-EE-CCSD	CCSD		EE_STATES
			EE_SNGLETS, EE_TRIPLETS
EOM-SF-CCSD	CCSD		SF_STATES
EOM-IP-CCSD	CCSD		IP_STATES
EOM-EA-CCSD	CCSD		EA_STATES
EOM-DIP-CCSD	CCSD		DIP_STATES
			DIP_SNGLETS, DIP_TRIPLETS
EOM-2SF-CCSD	CCSD	SDT or DT	DSF_STATES
EOM-EE-(2,3)	CCSD	SDT	EE_STATES
			EE_SNGLETS, EE_TRIPLETS
EOM-SF-(2,3)	CCSD	SDT	SF_STATES
EOM-IP-(2,3)	CCSD	SDT	IP_STATES
EOM-SF-CCSD(dT)	CCSD	SD(dT)	SF_STATES
EOM-SF-CCSD(fT)	CCSD	SD(fT)	SF_STATES
EOM-IP-CCSD(dT)	CCSD	SD(dT)	IP_STATES
EOM-IP-CCSD(fT)	CCSD	SD(fT)	IP_STATES
EOM-IP-CCSD(sT)	CCSD	SD(sT)	IP_STATES

Table 7.2: Commonly used EOM and CI models. ’SINGLETS’ and ’TRIPLETS’ are only available for closed-shell references.

The most relevant EOM-CC input options follow.

EE_STATES

Sets the number of excited state roots to find. For closed-shell reference, defaults into EE_SINGLETS. For open-shell references, specifies all low-lying states.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any excited states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ excited states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

EE_SINGLETS

Sets the number of singlet excited state roots to find. Valid only for closed-shell references.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any excited states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ excited states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

EE_TRIPLETS

Sets the number of triplet excited state roots to find. Valid only for closed-shell references.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any excited states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ excited states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

SF_STATES

Sets the number of spin-flip target states roots to find.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any excited states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ SF states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

DSF_STATES

Sets the number of doubly spin-flipped target states roots to find.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any DSF states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ doubly spin-flipped states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

IP_STATES

Sets the number of ionized target states roots to find. By default, $\beta$ electron will be removed (see EOM_IP_BETA).

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any IP states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ ionized states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

EOM_IP_ALPHA

Sets the number of ionized target states derived by removing $\alpha$ electron (M $_ s=-{{1}\over {2}}$ ).

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any IP/ $\alpha$ states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ ionized states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

EOM_IP_BETA

Sets the number of ionized target states derived by removing $\beta$ electron (M $_ s$ = ${{1}\over {2}}$ , default for EOM-IP).

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any IP/ $\beta$ states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ ionized states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

EA_STATES

Sets the number of attached target states roots to find. By default, $\alpha$ electron will be attached (see EOM_EA_ALPHA).

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any EA states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ EA states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

EOM_EA_ALPHA

Sets the number of attached target states derived by attaching $\alpha$ electron (M $_ s$ = ${{1}\over {2}}$ , default in EOM-EA).

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any EA states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ EA states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

EOM_EA_BETA

Sets the number of attached target states derived by attaching $\beta$ electron (M $_ s$ = $-{{1}\over {2}}$ , EA-SF).

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any EA states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ EA states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

DIP_STATES

Sets the number of DIP roots to find. For closed-shell reference, defaults into DIP_SINGLETS. For open-shell references, specifies all low-lying states.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any DIP states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ DIP states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

DIP_SINGLETS

Sets the number of singlet DIP roots to find. Valid only for closed-shell references.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any singlet DIP states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ DIP singlet states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

DIP_TRIPLETS

Sets the number of triplet DIP roots to find. Valid only for closed-shell references.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

0

Do not look for any DIP triplet states.

OPTIONS:

$[i,j,k\ldots ]$

Find $i$ DIP triplet states in the first irrep, $j$ states in the second irrep etc.

RECOMMENDATION:

None

Note: It is a symmetry of a transition rather than that of a target state which is specified in excited state calculations. The symmetry of the target state is a product of the symmetry of the reference state and the transition. For closed-shell molecules, the former is fully symmetric and the symmetry of the target state is the same as that of transition, however, for open-shell references this is not so.

Note: For the XX_STATES options, Q-Chem will increase the number of roots if it suspects degeneracy, or change it to a smaller value, if it cannot generate enough guess vectors to start the calculations.

EOM_FAKE_IPEA

If TRUE, calculates fake EOM-IP or EOM-EA energies and properties using the diffuse orbital trick. Default for EOM-EA and Dyson orbital calculations in CCMAN.

TYPE:

LOGICAL

DEFAULT:

FALSE (use proper EOM-IP code)

OPTIONS:

FALSE, TRUE

RECOMMENDATION:

None. This feature only works for CCMAN.

Note: When EOM_FAKE_IPEA is set to TRUE, it can change the convergence of Hartree-Fock iterations compared to the same job without EOM_FAKE_IPEA, because a very diffuse basis function is added to a center of symmetry before the Hartree-Fock iterations start. For the same reason, BASIS2 keyword is incompatible with EOM_FAKE_IPEA. In order to read Hartree-Fock guess from a previous job, you must specify EOM_FAKE_IPEA (even if you do not request for any correlation or excited states) in that previous job. Currently, the second moments of electron density and Mulliken charges and spin densities are incorrect for the EOM-IP/EA-CCSD target states.

EOM_USER_GUESS

Specifies if user-defined guess will be used in EOM calculations.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE

Solve for a state that has maximum overlap with a trans-n specified in $eom_user_guess.

RECOMMENDATION:

The orbitals are ordered by energy, as printed in the beginning of the CCMAN2 output. Not available in CCMAN.

EOM_SHIFT

Specifies energy shift in EOM calculations.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$n$

corresponds to $n\cdot 10^{-3}$ hartree shift (i.e., 11000 = 11 hartree); solve for eigenstates around this value.

RECOMMENDATION:

Not available in CCMAN.

EOM_NGUESS_DOUBLES

Specifies number of excited state guess vectors which are double excitations.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$n$

Include $n$ guess vectors that are double excitations

RECOMMENDATION:

This should be set to the expected number of doubly excited states, otherwise they may not be found.

EOM_NGUESS_SINGLES

Specifies number of excited state guess vectors that are single excitations.

TYPE:

INTEGER

DEFAULT:

Equal to the number of excited states requested

OPTIONS:

$n$

Include $n$ guess vectors that are single excitations

RECOMMENDATION:

Should be greater or equal than the number of excited states requested, unless .

EOM_PRECONV_SINGLES

When not zero, singly excited vectors are converged prior to a full excited states calculation. Sets the maximum number of iterations for pre-converging procedure.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0

do not pre-converge

1

pre-converge singles

RECOMMENDATION:

Sometimes helps with problematic convergence.

Note: In CCMAN, setting EOM_PRECONV_SINGLES = N would result in N Davidson iterations pre-converging singles.

EOM_PRECONV_DOUBLES

When not zero, doubly excited vectors are converged prior to a full excited states calculation. Sets the maximum number of iterations for pre-converging procedure

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0

Do not pre-converge

N

Perform N Davidson iterations pre-converging doubles.

RECOMMENDATION:

Occasionally necessary to ensure a doubly excited state is found. Also used in DSF calculations instead of EOM_PRECONV_SINGLES

Note: Not available in CCMAN2.

EOM_PRECONV_SD

When not zero, EOM vectors are pre-converged prior to a full excited states calculation. Sets the maximum number of iterations for pre-converging procedure.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0

do not pre-converge

N

perform N Davidson iterations pre-converging singles and doubles.

RECOMMENDATION:

Occasionally necessary to ensure that all low-lying states are found. Also, very useful in EOM(2,3) calculations.

None

Note: Not available in CCMAN2.

EOM_DAVIDSON_CONVERGENCE

Convergence criterion for the RMS residuals of excited state vectors.

TYPE:

INTEGER

DEFAULT:

5

Corresponding to $10^{-5}$

OPTIONS:

$n$

Corresponding to $10^{-n}$ convergence criterion

RECOMMENDATION:

Use the default. Normally this value be the same as EOM_DAVIDSON_THRESHOLD.

EOM_DAVIDSON_THRESHOLD

Specifies threshold for including a new expansion vector in the iterative Davidson diagonalization. Their norm must be above this threshold.

TYPE:

INTEGER

DEFAULT:

00103

Corresponding to 0.00001

OPTIONS:

$abcde$

Integer code is mapped to $abc\times 10^{-(de+2)}$ , i.e., 02505->2.5 $\times 10^{-6}$

RECOMMENDATION:

Use the default unless converge problems are encountered. Should normally be set to the same values as EOM_DAVIDSON_CONVERGENCE, if convergence problems arise try setting to a value slightly larger than EOM_DAVIDSON_CONVERGENCE.

EOM_DAVIDSON_MAXVECTORS

Specifies maximum number of vectors in the subspace for the Davidson diagonalization.

TYPE:

INTEGER

DEFAULT:

60

OPTIONS:

$n$

Up to $n$ vectors per root before the subspace is reset

RECOMMENDATION:

Larger values increase disk storage but accelerate and stabilize convergence.

EOM_DAVIDSON_MAX_ITER

Maximum number of iteration allowed for Davidson diagonalization procedure.

TYPE:

INTEGER

DEFAULT:

30

OPTIONS:

$n$

User-defined number of iterations

RECOMMENDATION:

Default is usually sufficient

EOM_IPEA_FILTER

If TRUE, filters the EOM-IP/EA amplitudes obtained using the diffuse orbital implementation (see EOM_FAKE_IPEA). Helps with convergence.

TYPE:

LOGICAL

DEFAULT:

FALSE (EOM-IP or EOM-EA amplitudes will not be filtered)

OPTIONS:

FALSE, TRUE

RECOMMENDATION:

None

Note: Not available in CCMAN2.

CC_FNO_THRESH

Initialize the FNO truncation and sets the threshold to be used for both cutoffs (OCCT and POVO).

TYPE:

INTEGER

DEFAULT:

None

OPTIONS:

range

0000-10000

$abcd$

Corresponding to $ab.cd$ %

RECOMMENDATION:

None

CC_FNO_USEPOP

Selection of the truncation scheme.

TYPE:

INTEGER

DEFAULT:

1

OCCT

OPTIONS:

0

POVO

RECOMMENDATION:

None

SCALE_NUCLEAR_CHARGE

Scales charge of each nuclei by a certain value. The nuclear repulsion energy is calculated for the unscaled nuclear charges.

TYPE:

INTEGER

DEFAULT:

0

No scaling.

OPTIONS:

$n$

A total positive charge of (1+ $n$ /100)e is added to the molecule.

RECOMMENDATION:

NONE

ADD_CHARGED_CAGE

Add a point charge cage of a given radius and total charge.

TYPE:

INTEGER

DEFAULT:

0

No cage.

OPTIONS:

0

No cage.

1

Dodecahedral cage.

2

Spherical cage.

RECOMMENDATION:

Spherical cage is expected to yield more accurate results, especially for small radii.

CAGE_RADIUS

Defines radius of the charged cage.

TYPE:

INTEGER

DEFAULT:

225

OPTIONS:

$n$

radius is $n$ /100 Å.

RECOMMENDATION:

None

CAGE_POINTS

Defines number of point charges for the spherical cage.

TYPE:

INTEGER

DEFAULT:

100

OPTIONS:

$n$

Number of point charges to use.

RECOMMENDATION:

None

CAGE_CHARGE

Defines the total charge of the cage.

TYPE:

INTEGER

DEFAULT:

400

Add a cage charged +4e.

OPTIONS:

$n$

Total charge of the cage is $n$ /100 a.u.

RECOMMENDATION:

None

7.7.14 Examples

Example 7.128 EOM-EE-OD and EOM-EE-CCSD calculations of the singlet excited states of formaldehyde

$molecule
   0 1
   O
   C  1  R1
   H  2  R2  1  A
   H  2  R2  1  A  3  180.
   
   R1  =  1.4
   R2  =  1.0
   A   =  120.
$end

$rem
   METHOD      eom-od
   BASIS       6-31+g
   EE_STATES   [2,2,2,2]
$end

@@@

$molecule
   read
$end

$rem
   METHOD        eom-ccsd
   BASIS         6-31+g
   EE_SINGLETS   [2,2,2,2]
   EE_TRIPLETS   [2,2,2,2]
$end

Example 7.129 EOM-EE-CCSD calculations of the singlet excited states of PYP using Cholesky decomposition

$molecule
0 1
...too long to enter...
$end

$rem
   METHOD          eom-ccsd
   BASIS           aug-cc-pVDZ
   PURECART        1112
   N_FROZEN_CORE   fc
   CC_T_CONV       4
   CC_E_CONV       6
   CHOLESKY_TOL    2    using CD/1e-2 threshold
   EE_SINGLETS     [2,2]
$end

Example 7.130 EOM-SF-CCSD calculations for methylene from high-spin $^3$ B $_2$ reference

$molecule
   0 3
   C
   H  1 rCH
   H  1 rCH 2 aHCH
   
   rCH   = 1.1167 
   aHCH  = 102.07
$end

$rem
   METHOD      eom-ccsd
   BASIS       6-31G*
   SCF_GUESS   core
   SF_STATES   [2,0,0,2]   Two singlet A1 states and singlet and triplet B2 states
$end

Example 7.131 EOM-SF-MP2 calculations for SiH $_2$ from high-spin $^3$ B $_2$ reference. Both energies and properties are computed.

$molecule
   0 3
   Si
   H  1 1.5145
   H  1 1.5145 2 92.68
$end

$rem
   BASIS            = cc-pVDZ       
   UNRESTRICTED     = true
   SCF_CONVERGENCE  = 8
   METHOD           = eom-mp2
   SF_STATES        = [1,1,0,0]
   CC_EOM_PROP_TE   = true   ! Compute <S^2> of excited states
$end

Example 7.132 EOM-IP-CCSD calculations for NO $_3$ using closed-shell anion reference

$molecule
   -1 1
   N
   O  1  r1
   O  1  r2   2 A2
   O  1  r2   2 A2    3 180.0
   
   r1    = 1.237
   r2    = 1.237
   A2    = 120.00
$end

$rem
   METHOD       eom-ccsd
   BASIS        6-31G*
   IP_STATES    [1,1,2,1]    ground and excited  states of the radical
$end

Example 7.133 EOM-IP-CCSD calculation using FNO with OCCT=99%.

$molecule
   0 1
   O
   H  1  1.0
   H  1  1.0  2  100.
$end

$rem
   METHOD          eom-ccsd
   BASIS           6-311+G(2df,2pd)
   IP_STATES       [1,0,1,1]
   CC_FNO_THRESH   9900        99% of the total natural population recovered
$end

Example 7.134 EOM-IP-MP2 calculation of the three low lying ionized states of the phenolate anion

$molecule
   0 1
   C   -0.189057  -1.215927  -0.000922
   H   -0.709319  -2.157526  -0.001587
   C    1.194584  -1.155381  -0.000067
   H    1.762373  -2.070036  -0.000230
   C    1.848872   0.069673   0.000936
   H    2.923593   0.111621   0.001593
   C    1.103041   1.238842   0.001235
   H    1.595604   2.196052   0.002078
   C   -0.283047   1.185547   0.000344
   H   -0.862269   2.095160   0.000376
   C   -0.929565  -0.042566  -0.000765
   O   -2.287040  -0.159171  -0.001759
   H   -2.663814   0.725029   0.001075
$end

$rem
   THRESH      16
   CC_MEMORY   30000
   BASIS       6-31+g(d)
   METHOD      eom-mp2
   IP_STATES   [3]
$end

Example 7.135 EOM-EE-MP2T calculation of the $H_2$ excitation energies

$molecule
   0 1
   H   0.0000   0.0000   0.0000
   H   0.0000   0.0000   0.7414
$end
    
$rem
   THRESH     16
   BASIS      cc-pvdz
   METHOD     eom-mp2t
   EE_STATES  [3,0,0,0,0,0,0,0]
$end

Example 7.136 EOM-EA-CCSD calculation of CN using user-specified guess

$molecule
   +1 1
   C
   N  1  1.1718
$end

$rem
   METHOD          = eom-ccsd
   BASIS          = 6-311+g*
   EA_STATES      = [1,1,1,1]
   CC_EOM_PROP    = true
   EOM_USER_GUESS = true   ! attach to HOMO, HOMO+1, and HOMO+3
$end

$eom_user_guess
   1  2  4
$end

Example 7.137 DSF-CIDT calculation of methylene starting with quintet reference

$molecule
   0 5
   C 
   H 1 CH
   H 1 CH 2 HCH

   CH  = 1.07
   HCH = 111.0
$end

$rem
   METHOD               cisdt
   BASIS                6-31G
   DSF_STATES           [0,2,2,0]
   EOM_NGUESS_SINGLES   0   
   EOM_NGUESS_DOUBLES   2
$end

Example 7.138 EOM-EA-CCSD job for cyano radical. We first do Hartree-Fock calculation for the cation in the basis set with one extremely diffuse orbital (EOM_FAKE_IPEA) and use these orbitals in the second job. We need make sure that the diffuse orbital is occupied using the OCCUPIED keyword. No SCF iterations are performed as the diffuse electron and the molecular core are uncoupled. The attached states show up as “excited” states in which electron is promoted from the diffuse orbital to the molecular ones.

$molecule
   +1 1
   C
   N 1 bond

   bond   1.1718
$end

$rem
   METHOD            hf
   BASIS             6-311+G*
   PURECART          111
   SCF_CONVERGENCE   8
   EOM_FAKE_IPEA     true
$end

@@@

$molecule
   0 2
   C
   N 1 bond
   
   bond   1.1718
$end

$rem
   BASIS            6-311+G*
   PURECART         111
   SCF_GUESS        read
   MAX_SCF_CYCLES   0
   METHOD           eom-ccsd
   CC_DOV_THRESH    2501   use thresh for CC iters with convergence problems
   EA_STATES        [2,0,0,0]
   EOM_FAKE_IPEA    true
$end

$occupied
   1 2 3 4 5 6 14
   1 2 3 4 5 6
$end

Example 7.139 DIP-EOM-CCSD calculation of methylene with charged cage stabilization.

$molecule
   -2 1
   C   0.000000     0.000000     0.106788
   H  -0.989216     0.000000    -0.320363
   H   0.989216     0.000000    -0.320363
$end

$rem
   BASIS                    = 6-311g(d,p)
   SCF_ALGORITHM            = diis_gdm
   SYMMETRY                 = false
   METHOD                   = eom-ccsd
   CC_SYMMETRY              = false
   DIP_SINGLETS             = [1]  ! Compute one EOM-DIP singlet state
   DIP_TRIPLETS             = [1]  ! Compute one EOM-DIP triplet state
   EOM_DAVIDSON_CONVERGENCE = 5
   CC_EOM_PROP              = true ! Compute excited state properties
   ADD_CHARGED_CAGE         = 2    ! Install a charged sphere around the molecule
   CAGE_RADIUS              = 225  ! Radius = 2.25 A
   CAGE_CHARGE              = 500  ! Charge = +5 a.u.
   CAGE_POINTS              = 100  ! Place 100 point charges
   CC_MEMORY                = 256  ! Use 256Mb of memory, increase for larger jobs
$end

Example 7.140 EOM-EE-CCSD calculation of excited states in NO $^-$ using scaled nuclear charge stabilization method.

$molecule
   -1 1
    N  -1.08735    0.0000    0.0000
    O   1.08735    0.0000    0.0000
$end

$rem
   INPUT_BOHR           = true
   BASIS                = 6-31g
   SYMMETRY             = false
   CC_SYMMETRY          = false
   METHOD               = eom-ccsd
   EE_SINGLETS          = [2]  ! Compute two EOM-EE singlet excited states
   EE_TRIPLETS          = [2]  ! Compute two EOM-EE triplet excited states
   CC_REF_PROP          = true ! Compute ground state properties
   CC_EOM_PROP          = true ! Compute excited state properties
   CC_MEMORY            = 256  ! Use 256Mb of memory, increase for larger jobs
   SCALE_NUCLEAR_CHARGE = 180  ! Adds +1.80e charge to the molecule
$end

Example 7.141 EOM-EE-CCSD calculation for phenol with user-specified guess requesting the EE transition from the occupied orbital number 24 (3 A") to the virtual orbital number 2 (23 A’)

$molecule
   0 1
   C    0.935445   -0.023376    0.000000
   C    0.262495    1.197399    0.000000
   C   -1.130915    1.215736    0.000000
   C   -1.854154    0.026814    0.000000
   C   -1.168805   -1.188579    0.000000
   C    0.220600   -1.220808    0.000000
   O    2.298632   -0.108788    0.000000
   H    2.681798    0.773704    0.000000
   H    0.823779    2.130309    0.000000
   H   -1.650336    2.170478    0.000000
   H   -2.939976    0.044987    0.000000
   H   -1.722580   -2.123864    0.000000
   H    0.768011   -2.158602    0.000000
$end

$rem
   METHOD                     EOM-CCSD
   BASIS                      6-31+G(d,p)
   CC_MEMORY                  3000   ccman2 memory
   MEM_STATIC                 250
   CC_T_CONV                  4      T-amplitudes convergence threshold
   CC_E_CONV                  6      Energy convergence threshold
   EE_STATES                  [0,1]  Calculate 1 A" states
   EOM_DAVIDSON_CONVERGENCE   5      Convergence threshold for the Davidson procedure
   EOM_USER_GUESS             true   Use user guess from $eom_user_guess section
$end

$eom_user_guess
   24
   2
$end

Example 7.142 Complex-scaled EOM-EE calculation for He. All roots of A $_ g$ symmetry are computed (full diagonalization)

$molecule
   0 1
   He  0    0  0.0
$end

$rem
   COMPLEX_CCMAN         1         engage complex_ccman
   METHOD                EOM-CCSD
   BASIS                 gen       use general basis
   PURECART              1111
   EE_SINGLETS           [2000,0,0,0,0,0,0,0]  compute all Ag excitations 
   EOM_DAVIDSON_CONV     5
   EOM_DAVIDSON_THRESH   5
   EOM_NGUESS_SINGLES    2000      Number of guess singles
   EOM_NGUESS_DOUBLES    2000      Number of guess doubles
   CC_MEMORY             2000
   MEM_TOTAL             3000
$end

$complex_ccman
   CS_HF 1                         Use complex HF
   CS_ALPHA 1000                   Set alpha equal 1
   CS_THETA 300                    Set theta (angle) equals 0.3 (radian)
$end

$basis
He   0
S    4    1.000000
   2.34000000E+02    2.58700000E-03 
   3.51600000E+01    1.95330000E-02 
   7.98900000E+00    9.09980000E-02 
   2.21200000E+00    2.72050000E-01 
S    1    1.000000
   6.66900000E-01    1.00000000E+00 
S    1    1.000000
   2.08900000E-01    1.00000000E+00 
P    1    1.000000
   3.04400000E+00    1.00000000E+00 
P    1    1.000000
   7.58000000E-01    1.00000000E+00 
D    1    1.000000
   1.96500000E+00    1.00000000E+00 
S    1    1.000000
   5.13800000E-02    1.00000000E+00 
P    1    1.000000
   1.99300000E-01    1.00000000E+00 
D    1    1.000000
   4.59200000E-01    1.00000000E+00 
S    1    1.000000
   2.44564000E-02    1.00000000E+00
S    1    1.000000
   1.2282000E-02    1.00000000E+00
S    1    1.000000
   6.1141000E-03    1.00000000E+00
P   1   1.0
    8.130000e-02    1.0
P   1   1.0
    4.065000e-02    1.0
P   1   1.0
    2.032500e-02    1.0
D   1   1.0
    2.34375e-01    1.0
D   1   1.0
    1.17187e-01    1.0
D   1   1.0
    5.85937e-02    1.0
****
$end

Example 7.143 CAP-augmented EOM-EA-CCSD calculation for N $_2^-$ . aug-cc-pVTZ basis augmented by the 3s3p3d diffuse functions placed in the COM. Two EA states are computed for CAP strength $\eta$ =0.002

$molecule
   0 1
   N   0.0  0.0 -0.54875676501
   N   0.0  0.0  0.54875676501
   Gh  0.0  0.0  0.0
$end

$rem
   COMPLEX_CCMAN   1             engage complex_ccman          
   METHOD          EOM-CCSD
   BASIS           gen           use general basis
   EA_STATES       [0,0,2,0,0,0,0,0]  compute electron attachment energies
   CC_MEMORY       5000          ccman2 memory
   MEM_TOTAL       2000
   CC_EOM_PROP     true          compute excited state properties
$end

$complex_ccman
   CS_HF           1     Use complex HF
   CAP_ETA         200   Set strength of CAP potential 0.002
   CAP_X           2760  Set length of the box along x dimension
   CAP_Y           2760  Set length of the box along y dimension
   CAP_Z           4880  Set length of the box along z dimension
   CAP_TYPE        1     Use cuboid CAP
$end

$basis
N    0
aug-cc-pvtz
****
Gh   0
S    1    1.000000
   2.88000000E-02    1.00000000E+00
S    1    1.000000
   1.44000000E-02    1.00000000E+00
S    1    1.000000
   0.72000000E-02    1.00000000E+00
P    1    1.000000
   2.45000000E-02    1.00000000E+00
P    1    1.000000
   1.22000000E-02    1.00000000E+00
P    1    1.000000
   0.61000000E-02    1.00000000E+00
D    1    1.000000
   0.755000000E-01   1.00000000E+00
D    1    1.000000
   0.377500000E-01   1.00000000E+00
D    1    1.000000
   0.188750000E-01   1.00000000E+00
****
$end

Example 7.144 CAP-EOM-EE calculation of water, with wave-function analysis of state and transition properties

$molecule
   0 1
   O   0.00000000     0.00000000     0.13594219
   H   0.00000000    -1.44761450    -1.07875060
   H   0.00000000     1.44761450    -1.07875060
$end

$rem
   METHOD                     eom-ccsd
   BASIS                      6-31G**
   CC_MEMORY                  2000
   MEM_TOTAL                  2500
   SCF_CONVERGENCE            12
   CC_CONVERGENCE             11
   EOM_DAVIDSON_CONVERGENCE   11
   CC_EOM_PROP                TRUE
   CC_FULLRESPONSE            FALSE
   CC_TRANS_PROP              TRUE
   COMPLEX_CCMAN              1
   EE_STATES                  [1,0,2,0]
   INPUT_BOHR                 TRUE
   ! WFA KEYWORDS
   STATE_ANALYSIS             true
   MOLDEN_FORMAT              true
   NTO_PAIRS                  4
   POP_MULLIKEN               true
$end

$complex_ccman
CS_HF           1
CAP_TYPE        1
CAP_ETA         10000
CAP_X           2000
CAP_Y           2500
CAP_Z           2500
$end

Example 7.145 Formaldehyde, calculating EOM-IP-CCSD-S(D) and EOM-IP-MP2-S(D) energies of 4 valence ionized states

$molecule
   0 1
   C 
   H  1  1.096135
   H  1  1.096135  2  116.191164
   O  1  1.207459  2  121.904418  3  -180.000000 0
$end

$rem
   METHOD       eom-ccsd-s(d)
   BASIS        6-31G*
   IP_STATES    [1,1,1,1] 
$end

@@@

$molecule 
   read
$end

$rem
   METHOD       eom-mp2-s(d)
   BASIS        6-31G*
   IP_STATES    [1,1,1,1] 
$end

Example 7.146 Formaldehyde, calculating EOM-EE-CCSD states with C-PCM method.

$molecule
   0 1
   O
   C,1,R1
   H,2,R2,1,A
   H,2,R2,1,A,3,180.
   
   R1 = 1.4
   R2 = 1.0
   A  = 120.
$end

$rem
   METHOD           eom-ccsd
   BASIS            cc-pvdz
   EE_STATES        [4]
   SOLVENT_METHOD   pcm  
$end    

$pcm
   theory          cpcm 
$end

$solvent
   dielectric      4.34
   dielectric_infi 1.829
$end

Example 7.147 NO $_2^-$ , calculating EOM-IP-CCSD states with C-PCM method.

$molecule
   -1 1
   N1
   O2 N1 RNO
   O3 N1 RNO O2 AONO

   RNO  = 1.305
   AONO = 106.7
$end

$rem
   METHOD           eom-ccsd
   BASIS            cc-pvdz
   IP_STATES        [2]
   SOLVENT_METHOD   pcm  
$end    

$pcm
   theory          cpcm 
$end

$solvent
   dielectric      4.34
   dielectric_infi 1.829
$end

7.7.15 Non-Hartree-Fock Orbitals in EOM Calculations

In cases of problematic open-shell references, e.g., strongly spin-contaminated doublet, triplet or quartet states, one may choose to use DFT orbitals. This can be achieved by first doing DFT calculation and then reading the orbitals and turning Hartree-Fock off (by setting SCF_GUESS = READ MAX_SCF_CYCLES = 0 in the CCMAN or CCMAN2 job). In CCMAN, a more convenient way is just to specify EXCHANGE, e.g., if EXCHANGE = B3LYP, B3LYP orbitals will be computed and used.

Note: Using non-HF exchange in CCMAN2 is not possible.

7.7.16 Analytic Gradients and Properties for the CCSD and EOM-XX-CCSD Methods

The coupled-cluster package in Q-Chem can calculate properties of target EOM states including permanent dipoles, static polarizabilities, $\ensuremath{\langle }S^2\ensuremath{\rangle }$ and $\ensuremath{\langle }R^2\ensuremath{\rangle }$ values, nuclear gradients (and geometry optimizations). The target state of interest is selected by CC_STATE_TO_OPT $rem, which specifies the symmetry and the number of the EOM state. In addition to state properties, calculations of various interstate properties are available (transition dipoles, two-photon absorption transition moments (and cross-sections), spin-orbit couplings).

Analytic gradients are available for the CCSD and all EOM-CCSD methods for both closed- and open-shell references (UHF and RHF only), including frozen core/virtual functionality[Levchenko et al.(2005)Levchenko, Wang, and Krylov] (see also Section 6.13). These calculations should be feasible whenever the corresponding single-point energy calculation is feasible.

Note: Gradients for ROHF and non-HF (e.g., B3LYP) orbitals are not yet available.

For the CCSD and EOM-CCSD wave functions, Q-Chem currently can calculate permanent and transition dipole moments, oscillator strengths, $\ensuremath{\langle }R^2\ensuremath{\rangle }$ (as well as XX, YY and ZZ components separately, which is useful for assigning different Rydberg states, e.g., $3p_ x$ vs. $3s$ , etc.), and the $\ensuremath{\langle }S^2\ensuremath{\rangle }$ values. Interface of the CCSD and EOM-CCSD codes with the NBO 5.0 package is also available. Furthermore, excited state analyses can be requested for EOM-CCSD excited states. For EOM-MP2, only state properties (dipole moments, $\ensuremath{\langle }R^2\ensuremath{\rangle }$ , $\ensuremath{\langle }S^2\ensuremath{\rangle }$ are available). Similar functionality is available for some EOM-OD and CI models (CCMAN only).

Analysis of the real- and complex-valued EOM-CC wave functions can also be performed; see Sections 7.7.24 and 11.2.6. NTO analysis for EOM-IP/EA/SF states is, obviously, only available for the transitions between the EOM states, so CC_STATE_TO_OPT keyword needs to be used, as in calculations of transition properties.

Users must be aware of the point group symmetry of the system being studied and also the symmetry of the excited (target) state of interest. It is possible to turn off the use of symmetry using the CC_SYMMETRY. If set to FALSE the molecule will be treated as having $C_1$ symmetry and all states will be of $A$ symmetry.

Q-Chem allows flexible control of interstate properties calculations, by using CC_TRANS_PROP rem or rem section $trans_prop: the user can request the transitions between all computed EOM target states and the reference state (CC_TRANS_PROP=1) or the calculations of all transition properties between all computed EOM target states (CC_TRANS_PROP=2). By default, the reference state is the CCSD reference. To compute transition properties relative to a particular EOM state, use CC_STATE_TO_OPT.

By default, only one-electron properties are computed. To activate calculations of two-electron properties, such as NACs, SOCs, 2PA, additional keywords should be activated, as described below.

The $trans_prop rem section allows the user to specify precisely which properties and for which pairs of states to computed. When $trans_prop section is present in the input, it disables CC_TRANS_PROP rem.

$trans_prop
state_list         ! Start a list of states
ee_singlets 1 1    ! state 1: EE singlet with irrep = 1 and istate = 1
ee_triplets 1 2    ! state 2: EE triplet with irrep = 1 and istate = 2
ref                ! state 3: Reference state (can be CC or MP2, but the latter NYI 
                   ! in transition prop driver)
end_list           ! End list
state_pair_list    ! Start to specify pairs of states, 
3 1                ! transition from state 3 to state 1 (known bug here: CC state 
                   ! needs to be 1st one)
3 2                ! transition from state 3 to state 2 (known bug here: CC state 
                   ! needs to be 1st one)
end_pairs          ! End list of pairs
calc nac           ! Compute NAC for all transition pairs listed before this keyword
state_list         ! Start another list of states (user is able to request multiple 
                   ! state lists for multiple tasks)
ref                ! reference state
ee_singlets 0 0    ! zero means all requested irreps/istate in $rem
end_list
calc dipole soc    ! Compute transition dipole and SOC
calc opdm_norm     ! Compute norm of transition OPDM

Notes about $trans_prop rem section:

calc computes properties for the first pair list (or state list) before it.
The pair list is optional: if there is no pair list, all possible combinations within the state list will be considered.
Options after calc include: nac, soc, dyson, 2pa, dipole, default, pcm, opdm_norm, wfa. Currently, only some of them are implemented.
$trans_prop control for CVS-EOM-CCSD properties is not yet implemented.

Note: $trans_prop section is a new feature and is still under development — use on your own risk. Eventually, this section will replace other controls and will become a default.

7.7.16.1 Transition moments and cross-sections for two-photon absorption within EOM-EE-CCSD

Calculation of transition moments and cross-sections for two-photon absorption for EOM-EE-CCSD wave functions is available in Q-Chem (CCMAN2 only). Both CCSD-EOM and EOM-EOM transitions can be computed. The formalism is described in Ref. Nanda:2015. This feature is available both for canonical and RI/CD implementations. Relevant keywords are CC_EOM_2PA (turns on the calculation, controls NTO calculation), CC_STATE_TO_OPT (used for EOM-EOM transitions); additional customization can be performed using the $2pa section.

The $2pa section is used to specify the range of frequency-pairs satisfying the resonance condition. If $2pa section is absent in the input, the transition moments are computed for 2 degenerate photons with total energy matching the excitation energy of each target EOM state (for CCSD-EOM) or each EOM-EOM energy difference (for EOM-EOM transitions): $2 h\nu =E_{ex}$

$2pa                   Non-degenerate resonant 2PA
N_2PA_POINTS 6         Number of frequency pairs
OMEGA_1 500000 10000   Scans 500 cm$^{-1}$ to 550 cm$^{-1}$ 
                       in steps of 10 cm$^{-1}$
$end

N_2PA_POINTS is the number of frequency pairs across the spectrum. The first value associated with OMEGA_1 is the frequency $\times 1000$ in cm $^{-1}$ at the start of the spectrum and the second value is the step size $\times 1000$ in cm $^{-1}$ . The frequency of the second photon at each step is determined within the code as the excitation energy minus OMEGA_1.

To gain insight into computed cross sections for 2PA, one can perform NTO analysis of the response one-particle density matrices [Nanda and Krylov(2017)]. To activate NTO analysis of the 2PA response one-particle transition density matrices, set STATE_ANALYSIS=TRUE, MOLDEN_FORMAT=TRUE (to export the orbitals as MOLDEN files), NTO_PAIRS (specifies the number of orbitals to print). The NTO analysis will be performed for the full 2PA response one-particle transition density matrices as well as the normalized $\omega$ DMs (see Ref. [Nanda and Krylov(2017)] for more details).

7.7.16.2 Calculations of Spin-Orbit Couplings Using EOM-CC Wave Functions

Calculations of spin-orbit couplings (SOCs) for EOM-CC wave functions is available in CCMAN2.[Epifanovsky et al.(2015)Epifanovsky, Klein, Stopkowicz, Gauss, and Krylov] We employ a perturbative approach in which SOCs are computed as matrix elements of the respective part of the Breit-Pauli Hamiltonian using zero-order non-relativistic wave functions. Both the full two-electron treatment and the mean-field approximation (a partial account of the two-electron contributions) are available for the EOM-EE/SF/IP/EA wave functions, as well as between the CCSD reference and EOM-EE/SF. To enable SOC calculation, transition properties between EOM states must be enabled via CC_TRANS_PROP, and SOC requested using CALC_SOC. By default, one-electron and mean-field two-electron couplings will be computed. Full two-electron coupling calculation is activated by setting CC_EOM_PROP_TE.

As with other EOM transition properties, the initial EOM state is set by CC_STATE_TO_OPT, and couplings are computed between that state and all other EOM states. In the absence of CC_STATE_TO_OPT, SOCs are computed between the reference state and all EOM-EE or EOM-SF states.

Note: In a spin-restricted case, such as EOM-EE calculations using closed-shell reference state, SOCs between the singlet and triplet EOM manifolds cannot be computed (only SOCs between the reference state and EOM triplets can be calculated). To compute SOCs between EOM-EE singlets and EOM-EE triplets, run the same job with UNRESTRICTED=TRUE, such that triplets and singlets appear in the same manifold.

7.7.16.3 Calculations of Non-Adiabatic Couplings Using EOM-CC Wave Functions

Calculations of non-adiabatic (derivative) couplings (NACs) for EOM-CC wave functions is available in CCMAN2. We employ Szalay’s approach in which couplings are computed by a modified analytic gradient code, via “summed states”:[Tajti and Szalay(2009)]

$\begin{equation} h^ x_{IJ}\equiv \langle \Psi _{I} | H^ x | \Psi _{J} \rangle = \frac{1}{2} \left( G_{(I+J)} - G_ I - G_ J \right) \protect \label{eq:Szalay:ExactH}, \end{equation}$

(7.54)

where, $G_ I$ , $G_ J$ , and $G_{IJ}$ are analytic gradients for states $I$ , $J$ , and a fictitious summed state $|\Psi _{I+J}\rangle \equiv |\Psi _ I\rangle + |\Psi _ J \rangle$ . Currently, NACs for EE/IP/EA are available.[Faraji et al.(2018)Faraji, Matsika, and Krylov]

Note: Note that the individual components of the NAC vector depend on the molecular orientation.

7.7.16.4 Calculations of Static Polarizabilities for CCSD and EOM-CCSD Wave Functions

Calculation of the static dipole polarizability for the CCSD and EOM-EE/SF wave function is available in CCMAN2. CCSD polarizabilities are calculated as second derivatives of the CCSD energy.[Nanda and Krylov(2016)] Only the response of the cluster amplitudes is taken into the account; orbital relaxation is not included. Currently, this feature is available for the canonical implementation only. Relevant keywords are CC_POL (turns on the calculation), EOM_POL (turns on the calculation for EOM states, otherwise, only the CCSD polarizability will be computed), and CC_REF_PROP/CC_FULLRESPONSE (both must be set to TRUE).

Note: EOM-CCSD polarizabilities are available for EE and SF wave functions only.

7.7.17 EOM-CC Optimization and Properties Job Control

CC_STATE_TO_OPT

Specifies which state to optimize (or from which state compute EOM-EOM inter-state properties).

TYPE:

INTEGER ARRAY

DEFAULT:

None

OPTIONS:

[ $i$ , $j$ ]

optimize the $j$ th state of the $i$ th irrep.

RECOMMENDATION:

None

Note: The state number should be smaller or equal to the number of excited states calculated in the corresponding irrep.

Note: If analytic gradients are not available, the finite difference calculations will be performed and the symmetry will be turned off. In this case, CC_STATE_TO_OPT should be specified assuming C $_1$ symmetry, i.e., as [1,N] where N is the number of state to optimize (the states are numbered from 1).

CC_EOM_PROP

Whether or not the non-relaxed (expectation value) one-particle EOM-CCSD target state properties will be calculated. The properties currently include permanent dipole moment, the second moments $\ensuremath{\langle }X^2\ensuremath{\rangle }$ , $\ensuremath{\langle }Y^2\ensuremath{\rangle }$ , and $\ensuremath{\langle }Z^2\ensuremath{\rangle }$ of electron density, and the total $\ensuremath{\langle }R^2\ensuremath{\rangle }= \ensuremath{\langle }X^2\ensuremath{\rangle }+\ensuremath{\langle }Y^2\ensuremath{\rangle }+\ensuremath{\langle }Z^2\ensuremath{\rangle }$ (in atomic units). Incompatible with JOBTYPE=FORCE, OPT, FREQ.

TYPE:

LOGICAL

DEFAULT:

FALSE (no one-particle properties will be calculated)

OPTIONS:

FALSE, TRUE

RECOMMENDATION:

Additional equations (EOM-CCSD equations for the left eigenvectors) need to be solved for properties, approximately doubling the cost of calculation for each irrep. The cost of the one-particle properties calculation itself is low. The one-particle density of an EOM-CCSD target state can be analyzed with NBO or libwfa packages by specifying the state with CC_STATE_TO_OPT and requesting NBO = TRUE and CC_EOM_PROP = TRUE.

CC_TRANS_PROP

Whether or not the transition dipole moment (in atomic units) and oscillator strength for the EOM-CCSD target states will be calculated. By default, the transition dipole moment is calculated between the CCSD reference and the EOM-CCSD target states. In order to calculate transition dipole moment between a set of EOM-CCSD states and another EOM-CCSD state, the CC_STATE_TO_OPT must be specified for this state.

TYPE:

INTEGER

DEFAULT:

0 (no transition properties will be calculated)

OPTIONS:

1 (calculate transition properties between all computed EOM state and the reference state)

2 (calculate transition properties between all pairs of EOM states)

RECOMMENDATION:

NONE

Additional equations (for the left EOM-CCSD eigenvectors plus lambda CCSD equations in case if transition properties between the CCSD reference and EOM-CCSD target states are requested) need to be solved for transition properties, approximately doubling the computational cost. The cost of the transition properties calculation itself is low.

Note: When $trans_prop section is present in the input, it disables CC_TRANS_PROP rem.

CC_EOM_2PA

Whether or not the transition moments and cross-sections for two-photon absorption will be calculated. By default, the transition moments are calculated between the CCSD reference and the EOM-CCSD target states. In order to calculate transition moments between a set of EOM-CCSD states and another EOM-CCSD state, the CC_STATE_TO_OPT must be specified for this state. If 2PA NTO analysis is requested, the CC_EOM_2PA value is redundant as long as CC_EOM_2PA $> 0$ .

TYPE:

INTEGER

DEFAULT:

0 (do not compute 2PA transition moments)

OPTIONS:

1

Compute 2PA using the fastest algorithm (use $\tilde{\sigma }$ -intermediates for canonical

and $\sigma$ -intermediates for RI/CD response calculations).

2

Use $\sigma$ -intermediates for 2PA response equation calculations.

3

Use $\tilde{\sigma }$ -intermediates for 2PA response equation calculations.

RECOMMENDATION:

Additional response equations (6 for each target state) will be solved, which increases the cost of calculations. The cost of 2PA moments is about 10 times that of energy calculation. Use the default algorithm. Setting CC_EOM_2PA $> 0$ turns on CC_TRANS_PROP.

CALC_SOC

Whether or not the spin-orbit couplings between CC/EOM/ADC/CIS/TDDFT electronic states will be calculated. In the CC/EOM-CC suite, by default the couplings are calculated between the CCSD reference and the EOM-CCSD target states. In order to calculate couplings between EOM states, CC_STATE_TO_OPT must specify the initial EOM state.

TYPE:

LOGICAL

DEFAULT:

FALSE (no spin-orbit couplings will be calculated)

OPTIONS:

FALSE, TRUE

RECOMMENDATION:

One-electron and mean-field two-electron SOCs will be computed by default. To enable full two-electron SOCs, two-particle EOM properties must be turned on (see CC_EOM_PROP_TE).

CALC_NAC

Whether or not non-adiabatic couplings will be calculated for the EOM-CC, CIS, and TDDFT wave functions.

TYPE:

INTEGER

DEFAULT:

0 (do not compute NAC)

OPTIONS:

1

NYI for EOM-CC

2

Compute NACs using Szalay’s approach (this what needs to be specified for EOM-CC).

RECOMMENDATION:

Additional response equations will be solved and gradients for all EOM states and for summed states will be computed, which increases the cost of calculations. Request only when needed and do not ask for too many EOM states.

CC_POL

Whether or not the static polarizability for the CCSD wave function will be calculated.

TYPE:

LOGICAL

DEFAULT:

FALSE (CCSD static polarizability will not be calculated)

OPTIONS:

FALSE, TRUE

RECOMMENDATION:

Static polarizabilities are expensive since they require solving three additional response equations. Do no request this property unless you need it.

EOM_POL

Whether or not the static polarizability for the EOM-CCSD wave function will be calculated.

TYPE:

LOGICAL

DEFAULT:

FALSE (EOM polarizability will not be calculated)

OPTIONS:

FALSE, TRUE

RECOMMENDATION:

Static polarizabilities are expensive since they require solving three additional response equations. Do no request this property unless you need it.

EOM_REF_PROP_TE

Request for calculation of non-relaxed two-particle EOM-CC properties. The two-particle properties currently include $\ensuremath{\langle }S^2\ensuremath{\rangle }$ . The one-particle properties also will be calculated, since the additional cost of the one-particle properties calculation is inferior compared to the cost of $\ensuremath{\langle }S^2\ensuremath{\rangle }$ . The variable CC_EOM_PROP must be also set to TRUE. Alternatively, CC_CALC_SSQ can be used to request $\ensuremath{\langle }S^2\ensuremath{\rangle }$ calculation.

TYPE:

LOGICAL

DEFAULT:

FALSE

(no two-particle properties will be calculated)

OPTIONS:

FALSE, TRUE

RECOMMENDATION:

The two-particle properties are computationally expensive since they require calculation and use of the two-particle density matrix (the cost is approximately the same as the cost of an analytic gradient calculation). Do not request the two-particle properties unless you really need them.

CC_FULLRESPONSE

Fully relaxed properties (including orbital relaxation terms) will be computed. The variable CC_EOM_PROP must be also set to TRUE.

TYPE:

LOGICAL

DEFAULT:

FALSE

(no orbital response will be calculated)

OPTIONS:

FALSE, TRUE

RECOMMENDATION:

Not available for non-UHF/RHF references. Only available for EOM/CI methods for which analytic gradients are available.

CC_SYMMETRY

Controls the use of symmetry in coupled-cluster calculations

TYPE:

LOGICAL

DEFAULT:

TRUE

OPTIONS:

TRUE

Use the point group symmetry of the molecule

FALSE

Do not use point group symmetry (all states will be of $A$ symmetry).

RECOMMENDATION:

It is automatically turned off for any finite difference calculations, e.g. second derivatives.

STATE_ANALYSIS

Activates excited state analyses using libwfa.

TYPE:

LOGICAL

DEFAULT:

FALSE (no excited state analyses)

OPTIONS:

TRUE, FALSE

RECOMMENDATION:

Set to TRUE if excited state analysis is required, but also if plots of densities or orbitals are needed. For details see section 11.2.6.

7.7.17.1 Examples

Example 7.148 Geometry optimization for the excited open-shell singlet state, $^1B_2$ , of methylene followed by the calculations of the fully relaxed one-electron properties using EOM-EE-CCSD

$molecule
   0 1
   C
   H  1 rCH
   H  1 rCH 2 aHCH
   
   rCH    = 1.083
   aHCH   = 145.
$end

$rem
   JOBTYPE                   OPT
   METHOD                    EOM-CCSD
   BASIS                     cc-pVTZ
   SCF_GUESS                 CORE
   SCF_CONVERGENCE           9
   EE_SINGLETS               [0,0,0,1]
   EOM_NGUESS_SINGLES        2
   CC_STATE_TO_OPT           [4,1]
   EOM_DAVIDSON_CONVERGENCE  9    use tighter convergence for EOM amplitudes
$end

@@@

$molecule
   read
$end

$rem
   METHOD               EOM-CCSD
   BASIS                cc-pVTZ
   SCF_GUESS            READ
   EE_SINGLETS          [0,0,0,1]
   EOM_NGUESS_SINGLES   2
   CC_EOM_PROP          1  calculate properties for EOM states
   CC_FULLRESPONSE      1  use fully relaxed properties
$end

Example 7.149 Property and transition property calculation on the lowest singlet state of CH $_2$ using EOM-SF-CCSD

$molecule
   0 3
   C
   H  1 rch
   H  1 rch 2 ahch

  rch  = 1.1167 
  ahch = 102.07
$end

$rem
   METHOD             eom-ccsd
   BASIS              cc-pvtz
   SCF_GUESS          core
   SCF_CONVERGENCE    9
   SF_STATES          [2,0,0,3]   Get three 1^B2 and two 1^A1 SF states
   CC_EOM_PROP        1
   CC_TRANS_PROP      1
   CC_STATE_TO_OPT    [4,1] First EOM state in the 4th irrep 
$end

Example 7.150 Geometry optimization with tight convergence for the $^2$ A $_1$ excited state of CH $_2$ Cl, followed by calculation of non-relaxed and fully relaxed permanent dipole moment and $\langle S^2\rangle$ .

$molecule
   0 2
   H
   C   1  CH
   CL  2  CCL  1  CCLH
   H   2  CH   3  CCLH  1  DIH
   
   CH   = 1.096247
   CCL  = 2.158212
   CCLH = 122.0
   DIH  = 180.0
$end

$rem
   JOBTYPE                    OPT
   METHOD                     EOM-CCSD
   BASIS                      6-31G*  Basis Set
   SCF_GUESS                  SAD
   EOM_DAVIDSON_CONVERGENCE   9    EOM amplitude convergence
   CC_T_CONV                  9    CCSD amplitudes convergence
   EE_STATES                  [0,0,0,1]
   CC_STATE_TO_OPT            [4,1]
   EOM_NGUESS_SINGLES         2
   GEOM_OPT_TOL_GRADIENT      2
   GEOM_OPT_TOL_DISPLACEMENT  2
   GEOM_OPT_TOL_ENERGY        2
$end

@@@

$molecule
   read
$end

$rem
   METHOD             EOM-CCSD
   BASIS              6-31G*  Basis Set
   SCF_GUESS          READ
   EE_STATES          [0,0,0,1]
   EOM_NGUESS_SINGLES  2
   CC_EOM_PROP        1   calculate one-electron properties
   CC_EOM_PROP_TE     1   and two-electron properties (S^2)
$end

@@@

$molecule
   read
$end

$rem
   METHOD              EOM-CCSD
   BASIS               6-31G*  Basis Set
   SCF_GUESS           READ
   EE_STATES           [0,0,0,1]
   EOM_NGUESS_SINGLES  2
   CC_EOM_PROP         1  calculate one-electron properties
   CC_EOM_PROP_TE      1  and two-electron properties (S^2)CC_EXSTATES_PROP 1
   CC_FULLRESPONSE     1  same as above, but do fully relaxed properties
$end

Example 7.151 CCSD calculation on three $A_2$ and one $B_2$ state of formaldehyde. Transition properties will be calculated between the third $A_2$ state and all other EOM states

$molecule
   0  1
   O
   C  1  1.4
   H  2  1.0  1  120
   H  2  1.0  1  120  3 180
$end

$rem
   BASIS             6-31+G
   METHOD            EOM-CCSD
   EE_STATES         [0,3,0,1]
   CC_STATE_TO_OPT   [2,3]
   CC_TRANS_PROP     true
$end

Example 7.152 EOM-IP-CCSD geometry optimization of X $^2B_2$ state of $\rm H_2O^+$ .

$molecule
   0 1
   H    0.774767     0.000000     0.458565
   O    0.000000     0.000000    -0.114641
   H   -0.774767     0.000000     0.458565
$end

$rem
   JOBTYPE           opt
   METHOD            eom-ccsd
   BASIS             6-311G
   IP_STATES         [0,0,0,1]
   CC_STATE_TO_OPT   [4,1]
$end

Example 7.153 CAP-EOM-EA-CCSD geometry optimization of the $^2B_1$ anionic resonance state of formaldehyde. The applied basis is aug-cc-pVDZ augmented by 3s3p diffuse functions on heavy atoms.

$molecule
   0 1
   C       0.0000000000     0.0000000000     0.5721328608
   O       0.0000000000     0.0000000000    -0.7102635035
   H       0.9478180646     0.0000000000     1.1819748108
   H      -0.9478180646     0.0000000000     1.1819748108
$end

$rem
   JOBTYPE                    opt
   METHOD                     eom-ccsd
   BASIS                      gen
   SCF_CONVERGENCE            9
   CC_CONVERGENCE             9
   EOM_DAVIDSON_CONVERGENCE   9
   EA_STATES                  [0,0,0,2]
   CC_STATE_TO_OPT            [4,1]
   XC_GRID                    000250000974
   COMPLEX_CCMAN              1
$end

$complex_ccman
   CS_HF 1
   CAP_TYPE 1
   CAP_ETA 60
   CAP_X 3850
   CAP_Y 2950
   CAP_Z 6100
$end

$basis
 H   0
 S   3  1.00
       13.0100000         0.196850000E-01
       1.96200000         0.137977000
      0.444600000         0.478148000
 S   1  1.00
      0.122000000          1.00000000
 P   1  1.00
      0.727000000          1.00000000
 S   1  1.00
      0.297400000E-01      1.00000000
 P   1  1.00
      0.141000000          1.00000000
 ****
 C   0
 S   8  1.00
       6665.00000         0.692000000E-03
       1000.00000         0.532900000E-02
       228.000000         0.270770000E-01
       64.7100000         0.101718000
       21.0600000         0.274740000
       7.49500000         0.448564000
       2.79700000         0.285074000
      0.521500000         0.152040000E-01
 S   8  1.00
       6665.00000        -0.146000000E-03
       1000.00000        -0.115400000E-02
       228.000000        -0.572500000E-02
       64.7100000        -0.233120000E-01
       21.0600000        -0.639550000E-01
       7.49500000        -0.149981000
       2.79700000        -0.127262000
      0.521500000         0.544529000
 S   1  1.00
      0.159600000          1.00000000
 P   3  1.00
       9.43900000         0.381090000E-01
       2.00200000         0.209480000
      0.545600000         0.508557000
 P   1  1.00
      0.151700000          1.00000000
 D   1  1.00
      0.550000000          1.00000000
 S   1  1.00
      0.469000000E-01     1.00000000
 P   1  1.00
      0.404100000E-01     1.00000000
 D   1  1.00
      0.151000000         1.00000000
 S   1  1.00
      0.234500000E-01      1.00000000
 S   1  1.00
      0.117250000E-01      1.00000000
 S   1  1.00
      0.058625000E-01      1.00000000
 P   1  1.00
      0.202050000E-01      1.00000000
 P   1  1.00
      0.101025000E-01      1.00000000
 P   1  1.00
      0.050512500E-01      1.00000000
 ****
 O   0
 S   8  1.00
       11720.0000         0.710000000E-03
       1759.00000         0.547000000E-02
       400.800000         0.278370000E-01
       113.700000         0.104800000
       37.0300000         0.283062000
       13.2700000         0.448719000
       5.02500000         0.270952000
       1.01300000         0.154580000E-01
 S   8  1.00
       11720.0000        -0.160000000E-03
       1759.00000        -0.126300000E-02
       400.800000        -0.626700000E-02
       113.700000        -0.257160000E-01
       37.0300000        -0.709240000E-01
       13.2700000        -0.165411000
       5.02500000        -0.116955000
       1.01300000         0.557368000
 S   1  1.00
      0.302300000          1.00000000
 P   3  1.00
       17.7000000         0.430180000E-01
       3.85400000         0.228913000
       1.04600000         0.508728000
 P   1  1.00
      0.275300000          1.00000000
 D   1  1.00
       1.18500000          1.00000000
 S   1  1.00
      0.789600000E-01      1.00000000
 P   1  1.00
      0.685600000E-01      1.00000000
 D   1  1.00
      0.332000000          1.00000000
 S   1  1.00
      0.394800000E-01      1.00000000
 S   1  1.00
      0.197400000E-01      1.00000000
 S   1  1.00
      0.098700000E-01      1.00000000
 P   1  1.00
      0.342800000E-01      1.00000000
 P   1  1.00
      0.171400000E-01      1.00000000
 P   1  1.00
      0.085700000E-01      1.00000000
 ****
$end

Example 7.154 Calculating resonant 2PA with degenerate photons.

$molecule
   0 1
   O
   H  1 0.959
   H  1 0.959 2 104.654
$end

$rem
   METHOD         eom-ccsd
   BASIS          aug-cc-pvtz
   EE_SINGLETS    [1,0,0,0]   1A_1 state
   CC_TRANS_PROP  1    Compute transition properties
   CC_EOM_2PA     1    Calculate 2PA cross-sections using the fastest algorithm
$end

Example 7.155 Non-degenerate, resonant 2PA scan over a range of frequency pairs.

$molecule
   0 1
   O
   H  1 0.959
   H  1 0.959 2 104.654
$end

$rem
   METHOD         eom-ccsd
   BASIS          aug-cc-pvtz
   EE_SINGLETS    [2,0,0,0]  Two A_1 states
   CC_TRANS_PROP  1          Calculate transition properties
   CC_EOM_2PA     1          Calculate 2PA cross-sections using the fastest algorithm
$end

$2pa               
   n_2pa_points 11 
   omega_1 500000 5000
$end

Example 7.156 Resonant 2PA with degenerate photons between two excited states.

$molecule
   0 1
   O
   H  1 0.959
   H  1 0.959 2 104.654
$end

$rem
   METHOD          eom-ccsd
   BASIS           aug-cc-pvtz
   EE_SINGLETS     [2,0,0,0] Two A_1 states
   CC_STATE_TO_OPT [1,1]     "Reference" state for transition properties is 1A_1 state 
   CC_TRANS_PROP   1         Compute transition properties
   CC_EOM_2PA      1         Calculate 2PA cross-sections using the fastest algorithm
$end

Example 7.157 Computation of spin-orbit couplings between closed-shell singlet and $M_ S=1$ triplet state in NH using EOM-SF-CCSD

$molecule
   0 3
   N
   H N 1.0450
$end

$rem
   METHOD          = eom-ccsd
   BASIS           = 6-31g
   SF_STATES       = [1,2,0,0]
   CC_TRANS_PROP   = true
   CALC_SOC        = true
   CC_STATE_TO_OPT = [1,1]
$end

Example 7.158 Computation of non-adiabatic couplings between EOM-EE states within triplet (first job) and singlet (second job) manifolds

$molecule
   +1 1
   H          0.00000        0.00000        0.0 
   He         0.00000        0.00000        3.0
$end

$rem
   JOBTYPE                  = FORCE
   BASIS                    = cc-pVDZ
   METHOD                   = EOM-CCSD
   INPUT_BOHR               = true
   EE_TRIPLETS              = [2]
   cc_eom_prop              = true    
   SYM_IGNORE               = true  Do not reorient molecule and turn off symmetry
   CALC_NAC                 = 2     Invoke Szalay NAC
   eom_davidson_convergence = 9     tight davidson convergence
   scf_convergence          = 9     Hartree-Fock convergence threshold 1e-9
   cc_convergence           = 9
$end

@@@

$molecule
   read
$end

$rem
   JOBTYPE                  = FORCE
   BASIS                    = cc-pVDZ
   METHOD                   = EOM-CCSD
   INPUT_BOHR               = true
   EE_STATES                = [2]   singlets
   SYM_IGNORE               = true  Do not reorient molecule and turn off symmetry
   CALC_NAC                 = 2     Invoke Szalay NAC
   eom_davidson_convergence = 9     tight davidson convergence
   scf_convergence          = 9     Hartree-Fock convergence threshold 1e-9
   cc_convergence           = 9
$end

Example 7.159 Calculation of the static dipole polarizability of the CCSD wave function of Helium.

$molecule
   0 1
   He
$end

$rem
   METHOD            ccsd
   BASIS             cc-pvdz
   CC_REF_PROP       1
   CC_POL            2
   CC_DIIS_SIZE      15
   CC_FULLRESPONSE   1
$end

7.7.18 EOM(2,3) Methods for Higher-Accuracy and Problematic Situations (CCMAN only)

In the EOM-CC(2,3) approach,[Hirata et al.(2000)Hirata, Nooijen, and Bartlett] the transformed Hamiltonian $\bar H$ is diagonalized in the basis of the reference, singly, doubly, and triply excited determinants, i.e., the excitation operator $R$ is truncated at triple excitations. The excitation operator $T$ , however, is truncated at double excitation level, and its amplitudes are found from the CCSD equations, just like for EOM-CCSD [or EOM-CC(2,2)] method.

The accuracy of the EOM-CC(2,3) method closely follows that of full EOM-CCSDT [which can be also called EOM-CC(3,3)], whereas computational cost of the former model is less.

The inclusion of triple excitations is necessary for achieving chemical accuracy (1 kcal/mol) for ground state properties. It is even more so for excited states. In particular, triple excitations are crucial for doubly excited states,[Hirata et al.(2000)Hirata, Nooijen, and Bartlett] excited states of some radicals and SF calculations (diradicals, triradicals, bond-breaking) when a reference open-shell state is heavily spin-contaminated. Accuracy of EOM-CCSD and EOM-CC(2,3) is compared in Table 7.7.18.

System	EOM-CCSD	EOM-CC(2,3)
Singly-excited electronic states	0.1–0.2 eV	0.01 eV
Doubly-excited electronic states	$\geq$ 1 eV	0.1–0.2 eV
Severe spin-contamination of the reference	$\sim$ 0.5 eV	$\leq$ 0.1 eV
Breaking single bond (EOM-SF)	0.1–0.2 eV	0.01 eV
Breaking double bond (EOM-2SF)	$\sim$ 1 eV	0.1–0.2 eV

Table 7.3: Performance of the EOM-CCSD and EOM-CC(2,3) methods

The applicability of the EOM-EE/SF-CC(2,3) models to larger systems can be extended by using their active-space variants, in which triple excitations are restricted to semi-internal ones.

Since the computational scaling of EOM-CC(2,3) method is ${\cal {O}}({N^8})$ , these calculations can be performed only for relatively small systems. Moderate size molecules (10 heavy atoms) can be tackled by either using the active space implementation or tiny basis sets. To achieve high accuracy for these systems, energy additivity schemes can be used. For example, one can extrapolate EOM-CCSDT/large basis set values by combining large basis set EOM-CCSD calculations with small basis set EOM-CCSDT ones.

Running the full EOM-CC(2,3) calculations is straightforward, however, the calculations are expensive with the bottlenecks being storage of the data on a hard drive and the CPU time. Calculations with around 80 basis functions are possible for a molecule consisting of four first row atoms (NO dimer). The number of basis functions can be larger for smaller systems.

Note: In EE calculations, one needs to always solve for at least one low-spin root in the first symmetry irrep in order to obtain the correlated EOM energy of the reference. The triples correction to the total reference energy must be used to evaluate EOM-(2,3) excitation energies.

Note: EOM-CC(2,3) works for EOM-EE, EOM-SF, and EOM-IP/EA. In EOM-IP, “triples” correspond to $3h2p$ excitations, and the computational scaling of EOM-IP-CC(2,3) is less.

7.7.19 Active-Space EOM-CC(2,3): Tricks of the Trade (CCMAN only)

Active space calculations are less demanding with respect to the size of a hard drive. The main bottlenecks here are the memory usage and the CPU time. Both arise due to the increased number of orbital blocks in the active space calculations. In the current implementation, each block can contain from 0 up to 16 orbitals of the same symmetry irrep, occupancy, and spin-symmetry. For example, for a typical molecule of C $_{\ensuremath{\mathrm{2v}}}$ symmetry, in a small/moderate basis set (e.g., TMM in 6-31G*), the number of blocks for each index is:

occupied: $(\alpha + \beta )\times ( a_1 + a_2 + b_1 + b_2) = 2\times 4 = 8$
virtuals: $(\alpha + \beta )\times ( 2 a_1 + a_2 + b_1 + 2 b_2) = 2\times 6 = 12$
(usually there are more than 16 $a_1$ and $b_2$ virtual orbitals).

In EOM-CCSD, the total number of blocks is $O^2V^2 = 8^2 \times 12^2 = 9216$ . In EOM-CC(2,3) the number of blocks in the EOM part is $O^3V^3 = 8^3 \times 12^3 = 884736$ . In active space EOM-CC(2,3), additional fragmentation of blocks occurs to distinguish between the restricted and active orbitals. For example, if the active space includes occupied and virtual orbitals of all symmetry irreps (this will be a very large active space), the number of occupied and virtual blocks for each index is 16 and 20, respectively, and the total number of blocks increases to $3.3\times 10^7$ . Not all of the blocks contain real information, some blocks are zero because of the spatial or spin-symmetry requirements. For the C $_{\ensuremath{\mathrm{2v}}}$ symmetry group, the number of non-zero blocks is about 10–12 times less than the total number of blocks, i.e., $3\times 10^6$ . This is the number of non-zero blocks in one vector. Davidson diagonalization procedure requires (2*MAX_VECTORS + 2*NROOTS) vectors, where MAX_VECTORS is the maximum number of vectors in the subspace, and NROOTS is the number of the roots to solve for. Taking NROOTS = 2 and MAX_VECTORS = 20, we obtain 44 vectors with the total number of non-zero blocks being $1.3\times 10^8$ .

In CCMAN implementation, each block is a logical unit of information. Along with real data, which are kept on a hard drive at all the times except of their direct usage, each non-zero block contains an auxiliary information about its size, structure, relative position with respect to other blocks, location on a hard drive, and so on. The auxiliary information about blocks is always kept in memory. Currently, the approximate size of this auxiliary information is about 400 bytes per block. It means, that in order to keep information about one vector ( $3\times 10^6$ blocks), 1.2 Gb of memory is required! The information about 44 vectors amounts 53 Gb. Moreover, the huge number of blocks significantly slows down the code.

To make the calculations of active space EOM-CC(2,3) feasible, we need to reduce the total number of blocks. One way to do this is to reduce the symmetry of the molecule to lower or C $_1$ symmetry group (of course, this will result in more expensive calculation). For example, lowering the symmetry group from C $_{\ensuremath{\mathrm{2v}}}$ to C $_{\ensuremath{\mathrm{s}}}$ would results in reducing the total number of blocks in active space EOM-CC(2,3) calculations in about $2^6 = 64$ times, and the number of non-zero blocks in about 30 times (the relative portion of non-zero blocks in C $_{\ensuremath{\mathrm{s}}}$ symmetry group is smaller compared to that in C $_{\ensuremath{\mathrm{2v}}}$ ).

Alternatively, one may keep the MAX_VECTORS and NROOTS parameters of Davidson’s diagonalization procedure as small as possible (this mainly concerns the MAX_VECTORS parameter). For example, specifying MAX_VECTORS = 12 instead of 20 would require 30% less memory.

One more trick concerns specifying the active space. In a desperate situation of a severe lack of memory, should the two previous options fail, one can try to modify (increase) the active space in such a way that the fragmentation of active and restricted orbitals would be less. For example, if there is one restricted occupied $b_1$ orbital and one active occupied $B_1$ orbital, adding the restricted $b_1$ to the active space will reduce the number of blocks, by the price of increasing the number of FLOPS. In principle, adding extra orbital to the active space should increase the accuracy of calculations, however, a special care should be taken about the (near) degenerate pairs of orbitals, which should be handled in the same way, i.e., both active or both restricted.

7.7.20 Job Control for EOM-CC(2,3)

EOM-CC(2,3) is invoked by METHOD=EOM-CC(2,3). The following options are available:

EOM_PRECONV_SD

Solves the EOM-CCSD equations, prints energies, then uses EOM-CCSD vectors as initial vectors in EOM-CC(2,3). Very convenient for calculations using energy additivity schemes.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$n$

Do $n$ SD iterations

RECOMMENDATION:

Turning this option on is recommended

CC_REST_AMPL

Forces the integrals, $T$ , and $R$ amplitudes to be determined in the full space even though the CC_REST_OCC and CC_REST_VIR keywords are used.

TYPE:

LOGICAL

DEFAULT:

TRUE

OPTIONS:

FALSE

Do apply restrictions

TRUE

Do not apply restrictions

RECOMMENDATION:

None

CC_REST_TRIPLES

Restricts $R_3$ amplitudes to the active space, i.e., one electron should be removed from the active occupied orbital and one electron should be added to the active virtual orbital.

TYPE:

INTEGER

DEFAULT:

1

OPTIONS:

1

Applies the restrictions

RECOMMENDATION:

None

CC_REST_OCC

Sets the number of restricted occupied orbitals including frozen occupied orbitals.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$n$

Restrict $n$ occupied orbitals.

RECOMMENDATION:

None

CC_REST_VIR

Sets the number of restricted virtual orbitals including frozen virtual orbitals.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$n$

Restrict $n$ virtual orbitals.

RECOMMENDATION:

None

To select the active space, orbitals can be reordered by specifying the new order in the $reorder_mosection. The section consists of two rows of numbers ( $\alpha$ and $\beta$ sets), starting from $1$ , and ending with $n$ , where $n$ is the number of the last orbital specified.

Example 7.160 Example $reorder_mo section with orbitals 16 and 17 swapped for both $\alpha$ and $\beta$ electrons

$reorder_mo
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 16
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 16
$end

7.7.20.1 Examples

Example 7.161 EOM-SF(2,3) calculations of methylene.

$molecule
   0 3
   C 
   H 1 CH
   H 1 CH 2 HCH
   
   CH  = 1.07
   HCH = 111.0
$end

$rem
   METHOD             eom-cc(2,3)
   BASIS              6-31G
   SF_STATES          [2,0,0,2]
   N_FROZEN_CORE      1
   N_FROZEN_VIRTUAL   1
   EOM_PRECONV_SD      20 Get EOM-CCSD energies first (max_iter=20).
$end

Example 7.162 This is active-space EOM-SF(2,3) calculations for methane with an elongated CC bond. HF MOs should be reordered as specified in the $reorder_mosection such that active space for triples consists of sigma and sigma* orbitals.

$molecule
   0 3
   C
   H  1  CH
   H  1  CHX  2  HCH
   H  1  CH   2  HCH  3  A120
   H  1  CH   2  HCH  4  A120
   
   CH   = 1.086
   HCH  = 109.4712206
   A120 = 120.
   CHX  = 1.8
$end

$rem
   METHOD            eom-cc(2,3)
   BASIS             6-31G*
   SF_STATES         [1,0]
   N_FROZEN_CORE     1   
   EOM_PRECONV_SD    20   does eom-ccsd first, max_iter=20
   CC_REST_TRIPLES   1   triples are restricted to the active space only
   CC_REST_AMPL      0   ccsd and eom singles and doubles are full-space 
   CC_REST_OCC       4   specifies active space
   CC_REST_VIR       17  specifies active space
   PRINT_ORBITALS    10  (number of virtuals to print)
$end

$reorder_mo
   1 2 5 4 3 
   1 2 3 4 5 
$end

Example 7.163 EOM-IP-CC(2,3) calculation of three lowest electronic states of water cation.

$molecule
0 1
   H   0.774767     0.000000     0.458565
   O   0.000000     0.000000    -0.114641
   H  -0.774767     0.000000     0.458565
$end

$rem
   METHOD       eom-cc(2,3)
   BASIS        6-311G
   IP_STATES    [1,0,1,1]
$end

7.7.21 Non-Iterative Triples Corrections to EOM-CCSD and CCSD

The effect of triple excitations to EOM-CCSD energies can be included via perturbation theory in an economical $N^7$ computational scheme. Using EOM-CCSD wave functions as zero-order wave functions, the second order triples correction to the $\mu$ th EOM-EE or SF state is:

$\begin{equation} \Delta E^{(2)}_{\mu } = - \frac{1}{36} \sum _{i,j,k} \sum _{a,b,c} \frac{ {\tilde\sigma _{ijk}^{abc}}(\mu ) {\sigma _{ijk}^{abc}}(\mu ) }{D_{ijk}^{abc} - \omega _{\mu }} \end{equation}$

(7.55)

where $i,j$ and $k$ denote occupied orbitals, and $a,b$ and $c$ are virtual orbital indices. $\omega _{\mu }$ is the EOM-CCSD excitation energy of the $\mu$ th state. The quantities $\tilde{\sigma }$ and $\sigma$ are:

	$\displaystyle \tilde{\sigma }_{ijk}^{abc}(\mu )$	$\displaystyle =$	$\displaystyle \langle \Phi _{0}\| ({L_{1}}_{\mu } + {L_{2}}_{\mu }) (H e^{(T_{1}+T_{2})})_{c} \| \Phi _{ijk}^{abc}\rangle$		(7.56)
	$\displaystyle \sigma _{ijk}^{abc}(\mu )$	$\displaystyle =$	$\displaystyle \langle \Phi _{ijk}^{abc}\|[H e^{(T_{1}+T_{2})} ({R_{0}}_{\mu } + {R_{1}}_{\mu } + {R_{2}}_{\mu })]_{c} \| \Phi _{0}\rangle$		(7.57)

where, the $L$ and $R$ are left and right eigen-vectors for $\mu$ th state. Two different choices of the denominator, $D_{ijk}^{abc}$ , define the (dT) and (fT) variants of the correction. In (fT), $D_{ijk}^{abc}$ is just Hartree-Fock orbital energy differences. A more accurate (but not fully orbital invariant) (dT) correction employs the complete three body diagonal of $\bar H$ , $\langle \Phi _{ijk}^{abc}| (He^{(T_{1}+T_{2})})_{C}|\Phi _{ijk}^{abc}\rangle$ , $D_{ijk}^{abc}$ as a denominator. For the reference (e.g., a ground-state CCSD wave function), the (fT) and (dT) corrections are identical to the CCSD(2) $_{T}$ and CR-CCSD(T) $_{L}$ corrections of Piecuch and coworkers.[Piecuch and Włoch(2005)]

The EOM-SF-CCSD(dT) and EOM-SF-CCSD(fT) methods yield a systematic improvement over EOM-SF-CCSD bringing the errors below 1 kcal/mol. For theoretical background and detailed benchmarks, see Ref. Manohar:2008.

Similar corrections are available for EOM-IP-CCSD,[Manohar et al.(2009)Manohar, Stanton, and Krylov] where triples correspond to $3h2p$ excitations and EOM-EA-CCSD, where triples correspond to $2h3p$ excitations.

Note: Due to the orbital non-invariance problem, using (dT) correction is discouraged.

Note: EOM-IP-CCSD(fT) correction is now available both in CCMAN and CCMAN2.

7.7.21.1 Job Control for Non-Iterative Triples Corrections

Triples corrections are requested by using METHOD or EOM_CORR:

METHOD

Specifies the calculation method.

TYPE:

STRING

DEFAULT:

No default value

OPTIONS:

EOM-CCSD(DT)

EOM-CCSD(dT), available for EE, SF, and IP

EOM-CCSD(FT)

EOM-CCSD(fT), available for EE, SF, IP, and EA

EOM-CCSD(ST)

EOM-CCSD(sT), available for IP

RECOMMENDATION:

None

EOM_CORR

Specifies the correlation level.

TYPE:

STRING

DEFAULT:

None

No correction will be computed

OPTIONS:

SD(DT)

EOM-CCSD(dT), available for EE, SF, and IP

SD(FT)

EOM-CCSD(fT), available for EE, SF, IP, and EA

SD(ST)

EOM-CCSD(sT), available for IP

RECOMMENDATION:

None

Note: In CCMAN2, EOM-IP-CCSD(fT) can be computed with or without USE_LIBPT=TRUE

7.7.21.2 Examples

Example 7.164 EOM-EE-CCSD(fT) calculation of CH $^{+}$

$molecule
   1 1
   C
   H  C CH

   CH  = 2.137130
$end

$rem
   INPUT_BOHR             true 
   METHOD                 eom-ccsd(ft)
   BASIS                  general
   EE_STATES              [1,0,1,1]
   EOM_DAVIDSON_MAX_ITER  60  increase number of Davidson iterations
$end

$basis
 H   0
 S   3  1.00
       19.24060000         0.3282800000E-01
       2.899200000         0.2312080000
      0.6534000000         0.8172380000
 S   1  1.00
      0.1776000000          1.000000000
 S   1  1.00
      0.0250000000          1.000000000
 P   1  1.00
       1.00000000          1.00000000
 ****
 C   0
 S   6  1.00
       4232.610000         0.2029000000E-02
       634.8820000         0.1553500000E-01
       146.0970000         0.7541100000E-01
       42.49740000         0.2571210000
       14.18920000         0.5965550000
       1.966600000         0.2425170000
 S   1  1.00
       5.147700000          1.000000000
 S   1  1.00
      0.4962000000          1.000000000
 S   1  1.00
      0.1533000000          1.000000000
 S   1  1.00
      0.0150000000          1.000000000
 P   4  1.00
       18.15570000         0.1853400000E-01
       3.986400000         0.1154420000
       1.142900000         0.3862060000
      0.3594000000         0.6400890000
 P   1  1.00
      0.1146000000          1.000000000
 P   1  1.00
      0.0110000000          1.000000000
 D   1  1.00
      0.750000000          1.00000000
 ****
$end

Example 7.165 EOM-SF-CCSD(dT) calculations of methylene

$molecule
   0 3
   C 
   H 1 CH
   H 1 CH 2 HCH
   
   CH  = 1.07
   HCH = 111.0
$end

$rem
   METHOD            eom-ccsd(dt)
   BASIS             6-31G
   SF_STATES         [2,0,0,2]
   N_FROZEN_CORE     1
   N_FROZEN_VIRTUAL  1
$end

Example 7.166 EOM-IP-CCSD(dT) calculations of Mg

$molecule
   0 1
   Mg    0.000000     0.000000    0.000000
$end

$rem
   JOBTYPE      sp
   METHOD       eom-ccsd(dt)
   BASIS        6-31g
   IP_STATES    [1,0,0,0,0,1,1,1]
$end

7.7.22 Potential Energy Surface Crossing Minimization

Potential energy surface crossing optimization procedure finds energy minima on crossing seams. On the seam, the potential surfaces are degenerated in the subspace perpendicular to the plane defined by two vectors: the gradient difference

$\begin{equation} {\rm \bf g} = \frac{\partial }{\partial {\rm \bf q}} (E_1 - E_2) \end{equation}$

(7.58)

and the derivative coupling

$\begin{equation} {\rm \bf h} = \left\langle \Psi _1 \left\vert \frac{\partial {\rm \bf H}}{\partial {\rm \bf q}} \right\vert \Psi _2 \right\rangle \end{equation}$

(7.59)

At this time Q-Chem is unable to locate crossing minima for states which have non-zero derivative coupling. Fortunately, often this is not the case. Minima on the seams of conical intersections of states of different multiplicity can be found as their derivative coupling is zero. Minima on the seams of intersections of states of different point group symmetry can be located as well.

To run a PES crossing minimization, CCSD and EOM-CCSD methods must be employed for the ground and excited state calculations respectively.

Note: MECP optimization is only available for methods with analytic gradients. Finite-difference evaluation of two gradients is not possible.

7.7.22.1 Job Control Options

XOPT_STATE_1, XOPT_STATE_2

Specify two electronic states the intersection of which will be searched.

TYPE:

[INTEGER, INTEGER, INTEGER]

DEFAULT:

No default value (the option must be specified to run this calculation)

OPTIONS:

[spin, irrep, state]

spin = 0

Addresses states with low spin,

see also EE_SINGLETS or IP_STATES,EA_STATES.

spin = 1

Addresses states with high spin,

see also EE_TRIPLETS.

irrep

Specifies the irreducible representation to which

the state belongs, for $C_{2v}$ point group symmetry

irrep = 1 for $A_1$ , irrep = 2 for $A_2$ ,

irrep = 3 for $B_1$ , irrep = 4 for $B_2$ .

state

Specifies the state number within the irreducible

representation, state = 1 means the lowest excited

state, state = 2 is the second excited state, etc..

0, 0, -1

Ground state.

RECOMMENDATION:

Only intersections of states with different spin or symmetry can be calculated at this time.

Note: The spin can only be specified when using closed-shell RHF references. In the case of open-shell references all states are treated together, see also EE_STATES. E.g., in SF calculations use spin=0 regardless of what is the actual multiplicity of the target state.

XOPT_SEAM_ONLY

Orders an intersection seam search only, no minimization is to perform.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE

Find a point on the intersection seam and stop.

FALSE

Perform a minimization of the intersection seam.

RECOMMENDATION:

In systems with a large number of degrees of freedom it might be useful to locate the seam first setting this option to TRUE and use that geometry as a starting point for the minimization.

7.7.22.2 Examples

Example 7.167 Minimize the intersection of Ã $^{1}$ B $_{2}$ and $\~{B}$ $^{1}$ A $_{2}$ states of the N $_{3}^{+}$ ion using EOM-CCSD method

$molecule
   1 1
   N1
   N2 N1 rnn
   N3 N2 rnn N1 annn

   rnn=1.46
   annn=70.0
$end

$rem
   JOBTYPE                opt
   METHOD                 eom-ccsd
   BASIS                  6-31g
   EE_SINGLETS            [0,2,0,2]   C2v point group symmetry
   XOPT_STATE_1           [0,4,1]     1B2 low spin state
   XOPT_STATE_2           [0,2,2]     2A2 low spin state
   XOPT_SEAM_ONLY         true        Find the seam only
   GEOM_OPT_TOL_GRADIENT  100
$end

$opt
CONSTRAINT                Set constraints on the N-N bond lengths
   stre  1  2  1.46
   stre  2  3  1.46
ENDCONSTRAINT
$end

@@@

$molecule
   READ
$end

$rem
   JOBTYPE                opt         Optimize the intersection seam
   METHOD                 eom-ccsd
   BASIS                  6-31g
   EE_SINGLETS            [0,2,0,2]
   XOPT_STATE_1           [0,4,1]
   XOPT_STATE_2           [0,2,2]
   GEOM_OPT_TOL_GRADIENT  30
$end

Example 7.168 Minimize the intersection of Ã $^{2}$ A $_{1}$ and $\~{B}$ $^{2}$ B $_{1}$ states of the NO $_{2}$ molecule using EOM-IP-CCSD method

$molecule
   -1  1
   N1
   O2  N1  rno
   O3  N1  rno  O2  aono

   rno  = 1.3040
   aono = 106.7
$end

$rem
   JOBTYPE                opt         Optimize the intersection seam
   UNRESTRICTED           true
   METHOD                 eom-ccsd
   BASIS                  6-31g
   IP_STATES              [1,0,1,0]   C2v point group symmetry
   EOM_FAKE_IPEA          1
   XOPT_STATE_1           [0,1,1]     1A1 low spin state
   XOPT_STATE_2           [0,3,1]     1B1 low spin state
   GEOM_OPT_TOL_GRADIENT  30          Tighten gradient tolerance
$END

7.7.23 Dyson Orbitals for Ionized or Attached States within the EOM-CCSD Formalism

Dyson orbitals can be used to compute total photodetachment/photoionization cross-sections, as well as angular distribution of photoelectrons. A Dyson orbital is the overlap between the N-electron molecular wave function and the N-1/N+1 electron wave function of the corresponding cation/anion:

	$\displaystyle \phi ^ d(1)$	$\displaystyle =$	$\displaystyle {{1}\over {N-1}}\int \Psi ^ N(1, \ldots , n) \Psi ^{N-1}(2, \ldots , n) d2 \ldots dn$		(7.60)
	$\displaystyle \phi ^ d(1)$	$\displaystyle =$	$\displaystyle {{1}\over {N+1}}\int \Psi ^ N(2, \ldots , n+1) \Psi ^{N+1}(1, \ldots , n+1) d2 \ldots d(n+1)$		(7.61)

For the Hartree-Fock wave functions and within Koopmans’ approximation, these are just the canonical HF orbitals. For correlated wave functions, Dyson orbitals are linear combinations of the reference molecular orbitals:

$\displaystyle \phi ^ d$	$\displaystyle =$	$\displaystyle \sum _ p \gamma _ p \phi _ p$	(7.62)
$\displaystyle \gamma _ p$	$\displaystyle =$	$\displaystyle \langle \Psi ^ N \| p^+ \| \Psi ^{N-1} \rangle$	(7.63)
$\displaystyle \gamma _ p$	$\displaystyle =$	$\displaystyle \langle \Psi ^ N \| p \| \Psi ^{N+1} \rangle$	(7.64)

The calculation of Dyson orbitals is straightforward within the EOM-IP/EA-CCSD methods, where cation/anion and initial molecule states are defined with respect to the same MO basis. Since the left and right CC vectors are not the same, one can define correspondingly two Dyson orbitals (left and right):

	$\displaystyle \gamma _ p^ R$	$\displaystyle =$	$\displaystyle \langle \Phi _0 e^{T_1+T_2} L^{EE} \|p^+\| R^{IP} e^{T_1+T_2} \Phi _0\rangle$		(7.65)
	$\displaystyle \gamma _ p^ L$	$\displaystyle =$	$\displaystyle \langle \Phi _0 e^{T_1+T_2} L^{IP} \|p\| R^{EE} e^{T_1+T_2} \Phi _0\rangle$		(7.66)

The norm of these orbitals is proportional to the one-electron character of the transition.

Dyson orbitals also offer qualitative insight visualizing the difference between molecular and ionized/attached states. In ionization/photodetachment processes, these orbitals can be also interpreted as the wave function of the leaving electron. For additional details, see Refs. Oana:2007,Oana:2009. Dyson orbitals can be used for computing total and differential photoelectron cross-sections using a stand-alone ezDyson code[Gozem and Krylov(2015)].

7.7.23.1 Dyson Orbitals Job Control

The calculation of Dyson orbitals is implemented for the ground (reference) and excited states ionization/electron attachment. To obtain the ground state Dyson orbitals one needs to run an EOM-IP/EA-CCSD calculation, request transition properties calculation by setting CC_TRANS_PROP=TRUE and CC_DO_DYSON = TRUE. The Dyson orbitals decomposition in the MO basis is printed in the output, for all transitions between the reference and all IP/EA states. At the end of the file, also the coefficients of the Dyson orbitals in the AO basis are available.

Two implementations of Dyson orbitals are currently available: (i) the original implementation in CCMAN; and (ii) new implementation in CCMAN2. The CCMAN implementation is using a diffuse orbital trick (i.e., EOM_FAKE_IPEA will be automatically set to TRUE in these calculations). Note: this implementation has a bug affecting the values of norms of Dyson orbitals (the shapes are correct); thus, using this code is strongly discouraged. The CCMAN2 implementation has all types of initial states available: Dyson orbitals from ground CC, excited EOM-EE, and spin-flip EOM-SF states; it is fully compatible with all helper features for EOM calculations, like FNO, RI, Cholesky decomposition. The CCMAN2 implementation can use a user-specified EOM guess (using EOM_USER_GUESS keyword and $eom_user_guess section), which is recommended for highly excited states (such as core-ionized states). In addition, CCMAN2 can calculate Dyson orbitals involving meta-stable states (see Section 7.7.6).

For calculating Dyson orbitals between excited or spin-flip states from the reference configuration and IP/EA states, same CC_TRANS_PROP = TRUE and CC_DO_DYSON = TRUE keywords have to be added to the combination of usual EOM-IP/EA-CCSD and EOM-EE-CCSD or EOM-SF-CCSD calculations (however, note separate keyword CC_DO_DYSON_EE = TRUE for CCMAN). The IP_STATES keyword is used to specify the target ionized states. The attached states are specified by EA_STATES. The EA-SF states are specified by EOM_EA_BETA. The excited (or spin-flipped) states are specified by EE_STATES and SF_STATES. The Dyson orbital decomposition in MO and AO bases is printed for each EE/SF-IP/EA pair of states first for reference, then for all excited states in the order: CC - IP/EA1, CC - IP/EA2, $\ldots$ , EE/SF1 - IP/EA1, EE/SF1 - IP/EA2, $\ldots$ , EE/SF2 - IP/EA1, EE/SF2 - IP/EA2, $\ldots$ , and so on. CCMAN implementation keeps reference transitions separate, in accordance with separating keywords.

CC_DO_DYSON

CCMAN2: starts all types of Dyson orbitals calculations. Desired type is determined by requesting corresponding EOM-XX transitions CCMAN: whether the reference-state Dyson orbitals will be calculated for EOM-IP/EA-CCSD calculations.

TYPE:

LOGICAL

DEFAULT:

FALSE (the option must be specified to run this calculation)

OPTIONS:

TRUE/FALSE

RECOMMENDATION:

none

CC_DO_DYSON_EE

Whether excited-state or spin-flip state Dyson orbitals will be calculated for EOM-IP/EA-CCSD calculations with CCMAN.

TYPE:

LOGICAL

DEFAULT:

FALSE (the option must be specified to run this calculation)

OPTIONS:

TRUE/FALSE

RECOMMENDATION:

none

Dyson orbitals are most easily visualized by setting GUI = 2 and reading the resulting checkpoint file into IQmol. In addition to the canonical orbitals, the Dyson orbitals will appear under the Surfaces item in the Model View. For step-by-step instructions, see ezDyson manual[Gozem and Krylov(2015)]. Alternatively Dyson orbitals can be plotted using IANLTY = 200 and the $plots utility. Only the sizes of the box need to be specified, followed by a line of zeros:

$plots
comment
   10   -2   2
   10   -2   2
   10   -2   2
   0     0   0    0
$plots

All Dyson orbitals on the Cartesian grid will be written in the resulting plot.mo file (only CCMAN). For RHF(UHF) reference, the columns order in plot.mo is: $\phi ^{lr}_1\alpha \ (\phi ^{lr}_1\beta ) \ \phi ^{rl}_1\alpha \ (\phi ^{rl}_1\beta ) \ \phi ^{lr}_2\alpha \ (\phi ^{lr}_2\beta ) \ \ldots$

In addition, setting the MAKE_CUBE_FILES keyword to TRUE will create cube files for Dyson orbitals which can be viewed with VMD or other programs (see Section 11.5.4 for details). This option is available for CCMAN and CCMAN2. The Dyson orbitals will be written to files mo.1.cube, mo.2.cube, $\ldots$ in the order $\phi ^{lr}_1 \ \phi ^{rl}_1 \ \ \phi ^{lr}_2 \ \phi ^{rl}_2 \ldots$ . For meta-stable states, the real and imaginary parts of the Dyson orbitals are written to separate files in the order $\text {Re}(\phi ^{lr}_1) \ \text {Re}(\phi ^{rl}_1) \ \ \text {Re}(\phi ^{lr}_2) \ \text {Re}(\phi ^{rl}_2) \ldots \text {Im}(\phi ^{lr}_1) \ \text {Im}(\phi ^{rl}_1) \ \ \text {Im}(\phi ^{lr}_2) \ \text {Im}(\phi ^{rl}_2) \ldots$

Visualization via the MolDen format is currently not available.

7.7.23.2 Examples

Example 7.169 Plotting grd-ex and ex-grd state Dyson orbitals for ionization of the oxygen molecule. The target states of the cation are $^2$ A $_ g$ and $^2$ B $_{2u}$ . Works for CCMAN only.

$molecule
   0 3
   O    0.000  0.000  0.000
   O    1.222  0.000  0.000
$end

$rem
   BASIS          6-31G*
   METHOD         eom-ccsd
   IP_STATES      [1,0,0,0,0,0,1,0] Target EOM-IP states
   CC_TRANS_PROP  true  request transition OPDMs to be calculated
   CC_DO_DYSON    true  calculate Dyson orbitals
   IANLTY         200
$end

$plots
plots excited states densities and trans densities
   10   -2   2
   10   -2   2
   10   -2   2
   0     0   0    0
$plots

Example 7.170 Plotting ex-ex state Dyson orbitals between the 1st $^2A_1$ excited state of the HO radical and the the 1st A $_1$ and A $_2$ excited states of HO $^-$ . Works for CCMAN only.

$molecule
   -1 1
   H    0.000   0.000  0.000
   O    1.000   0.000  0.000
$end

$rem
   METHOD             eom-ccsd
   BASIS              6-31G*
   IP_STATES          [1,0,0,0]  states of HO radical
   EE_STATES          [1,1,0,0]  excited states of HO- 
   CC_TRANS_PROP      true       calculate transition properties  
   CC_DO_DYSON_EE     true       calculate Dyson orbitals for ionization from ex. states 
   IANLTY             200                
$end

$plots
plot excited states densities and trans densities
   10   -2   2
   10   -2   2
   10   -2   2
   0     0   0    0
$plots

Example 7.171 Dyson orbitals for ionization of CO molecule; A $_1$ and B $_1$ ionized states requested.

$molecule
   0 1
   O
   C  O  1.131
$end

$rem
   CORRELATION           CCSD
   BASIS                 cc-pVDZ
   PURECART              111        5d, will be required for ezDyson
   IP_STATES             [1,0,1,0]  (A1,A2,B1,B2)
   CCMAN2                true
   CC_DO_DYSON           true
   CC_TRANS_PROP         true       necessary for Dyson orbitals job
   PRINT_GENERAL_BASIS   true       will be required for ezDyson
$end

Example 7.172 Dyson orbitals for ionization of H $_2$ O; core (A $_1$ ) state requested — ionization from O(1s).


$molecule
   0 1
   O
   H1  O  0.955
   H2  O  0.955  H1  104.5
$end

$rem
   CORRELATION           CCSD
   BASIS                 cc-pVTZ
   PURECART              111        5d, will be required for ezDyson
   IP_STATES             [1,0,0,0]  (A1,A2,B1,B2)
   EOM_USER_GUESS        1          on, further defined in $eom_user_guess
   CCMAN2                true
   CC_DO_DYSON           true
   CC_TRANS_PROP         true       necessary for Dyson orbitals job
   PRINT_GENERAL_BASIS   true       will be required for ezDyson
$end

$eom_user_guess
   1
$end

Example 7.173 Dyson orbitals for ionization of NO molecule using EOM-EA and a closed-shell cation reference; A $_1$ and B $_2$ states requested.

$molecule
   +1 1
   N   0.00000  0.00000  0.00000
   O   0.00000  0.00000  1.02286
$end

$rem
   CORRELATION            CCSD
   BASIS                  aug-cc-pVTZ
   PURECART               111       5d, will be required for ezDyson
   EA_STATES              [1,0,0,1] (A1,A2,B1,B2)
   CCMAN2                 true
   CC_DO_DYSON            true
   CC_TRANS_PROP          true      necessary for Dyson orbitals job
   PRINT_GENERAL_BASIS    true      will be required for ezDyson
$end

Example 7.174 Dyson orbitals for detachment from the meta-stable $^2\Pi _ g$ state of N $_2^-$ .

$molecule
   0 1
   N   0.0   0.0    0.55
   N   0.0   0.0   -0.55
   GH  0.0   0.0    0.0
$end

$rem
   METHOD            EOM-CCSD
   EA_STATES         [0,0,2,0,0,0,0,0]
   CC_MEMORY         5000
   MEM_STATIC        1000
   BASIS             GEN
   COMPLEX_CCMAN     TRUE
   CC_TRANS_PROP     TRUE
   CC_DO_DYSON       TRUE
   MAKE_CUBE_FILES   TRUE
   IANLTY            200
$end

$complex_ccman
   CS_HF             1
   CAP_TYPE          1
   CAP_X             2760
   CAP_Y             2760
   CAP_Z             4880
   CAP_ETA           400
$end

$plots
plot Dyson orbitals
   50 -10.0 10.0
   50 -10.0 10.0
   50 -10.0 10.0
   0 0 0 0
$end

$basis
N    0
S    8    1.000000
   1.14200000E+04    5.23000000E-04
   1.71200000E+03    4.04500000E-03
   3.89300000E+02    2.07750000E-02
   1.10000000E+02    8.07270000E-02
   3.55700000E+01    2.33074000E-01
   1.25400000E+01    4.33501000E-01
   4.64400000E+00    3.47472000E-01
   5.11800000E-01   -8.50800000E-03
S    8    1.000000
   1.14200000E+04   -1.15000000E-04
   1.71200000E+03   -8.95000000E-04
   3.89300000E+02   -4.62400000E-03
   1.10000000E+02   -1.85280000E-02
   3.55700000E+01   -5.73390000E-02
   1.25400000E+01   -1.32076000E-01
   4.64400000E+00   -1.72510000E-01
   5.11800000E-01    5.99944000E-01
S    1    1.000000
   1.29300000E+00    1.00000000E+00
S    1    1.000000
   1.78700000E-01    1.00000000E+00
P    3    1.000000
   2.66300000E+01    1.46700000E-02
   5.94800000E+00    9.17640000E-02
   1.74200000E+00    2.98683000E-01
P    1    1.000000
   5.55000000E-01    1.00000000E+00
P    1    1.000000
   1.72500000E-01    1.00000000E+00
D    1    1.000000
   1.65400000E+00    1.00000000E+00
D    1    1.000000
   4.69000000E-01    1.00000000E+00
F    1    1.000000
   1.09300000E+00    1.00000000E+00
S    1    1.000000
   5.76000000E-02    1.00000000E+00
P    1    1.000000
   4.91000000E-02    1.00000000E+00
D    1    1.000000
   1.51000000E-01    1.00000000E+00
F    1    1.000000
   3.64000000E-01    1.00000000E+00
****
GH   0
S    1    1.000000
   2.88000000E-02    1.00000000E+00
S    1    1.000000
   1.44000000E-02    1.00000000E+00
S    1    1.000000
   0.72000000E-02    1.00000000E+00
S    1    1.000000
   0.36000000E-02    1.00000000E+00
S    1    1.000000
   0.18000000E-02    1.00000000E+00
S    1    1.000000
   0.09000000E-02    1.00000000E+00
P    1    1.000000
   2.45000000E-02    1.00000000E+00
P    1    1.000000
   1.22000000E-02    1.00000000E+00
P    1    1.000000
   0.61000000E-02    1.00000000E+00
P    1    1.000000
   0.305000000E-02    1.00000000E+00
P    1    1.000000
   0.152500000E-02    1.00000000E+00
P    1    1.000000
   0.076250000E-02    1.00000000E+00
D    1    1.000000
   0.755000000E-01    1.00000000E+00
D    1    1.000000
   0.377500000E-01    1.00000000E+00
D    1    1.000000
   0.188750000E-01    1.00000000E+00
D    1    1.000000
   0.094375000E-01    1.00000000E+00
D    1    1.000000
   0.047187500E-01    1.00000000E+00
D    1    1.000000
   0.023593750E-01    1.00000000E+00
****
$end

Example 7.175 Dyson orbitals for ionization of triplet O $_2$ and O $_2^-$ at slightly stretched (relative to the equilibrium O $_2$ geometry); B $_{3g}$ states are requested.

$comment
   EOM-IP-CCSD/6-311+G* and EOM-EA-CCSD/6-311+G* levels of theory, 
   UHF reference.  Start from O2:
    1) detach electron - ionization of neutral (alpha IP).
    2) attach electron, use EOM-EA w.f. as initial state 
       - ionization of anion (beta EA).
$end 

$molecule
   0 3
   O   0.00000  0.00000  0.00000
   O   0.00000  0.00000  1.30000
$end

$rem
   CORRELATION           CCSD
   BASIS                 6-311(3+)G*
   PURECART              2222          6d, will be required for ezDyson
   EOM_IP_ALPHA          [0,0,0,1,0,0,0,0]  (Ag,B1g,B2g,B3g,Au,B1u,B2u,B3u)
   EOM_EA_BETA           [0,0,0,1,0,0,0,0]  (Ag,B1g,B2g,B3g,Au,B1u,B2u,B3u)
   CCMAN2                true
   CC_DO_DYSON           true
   CC_TRANS_PROP         true          necessary for Dyson orbitals job
   PRINT_GENERAL_BASIS   true          will be required for ezDyson
$end

Example 7.176 Dyson orbitals for ionization of formaldehyde from the first excited state AND from the ground state

$molecule
   0 1
   O     1.535338855      0.000000000     -0.438858006
   C     1.535331598     -0.000007025      0.767790994
   H     1.535342484      0.937663512      1.362651452
   H     1.535342484     -0.937656488      1.362672535
$end

$rem
   CORRELATION           CCSD
   BASIS                 6-31G*
   PURECART              2222    6d, will be required for ezDyson
   CCMAN2                true    new Dyson code
   EE_STATES             [1]
   EOM_IP_ALPHA          [1]
   EOM_IP_BETA           [1]
   CC_TRANS_PROP         true    necessary for Dyson orbitals job
   CC_DO_DYSON           true
   PRINT_GENERAL_BASIS   true    will be required for ezDyson
$end

Example 7.177 Dyson orbitals for core ionization of Li atom use Li $^+$ as a reference, get neutral atom via EOM-EA get 1st excitation for the cation via EOM-EE totally: core ionization AND 1st ionization of Li atom

$molecule
   +1 1
   Li   0.00000  0.00000  0.00000
$end

$rem
   CORRELATION            CCSD
   BASIS                  6-311+G*
   PURECART               2222          6d, will be required for ezDyson
   CCMAN2                 true          new Dyson code
   EE_STATES              [1,0,0,0,0,0,0,0]
   EA_STATES              [1,0,0,0,0,0,0,0]
   EOM_NGUESS_SINGLES     5             to converge to the lowest EA state
   CC_TRANS_PROP          true          necessary for Dyson orbitals job
   CC_DO_DYSON            true
   PRINT_GENERAL_BASIS    true          will be required for ezDyson
$end

Example 7.178 Dyson orbitals for ionization of CH2 from high-spin triplet reference and from the lowest SF state

$molecule
   0 3
   C
   H  1 rCH
   H  1 rCH 2 aHCH
   
   rCH    = 1.1167
   aHCH   = 102.07
$end

$rem
   CORRELATION           CCSD
   BASIS                 6-31G*
   SCF_GUESS             core
   CCMAN2                true    new Dyson code
   CC_SYMMETRY           false
   SF_STATES             [1]
   EOM_IP_ALPHA          [2]     one should be careful to request
   EOM_EA_BETA           [2]     meaningful spin for EA/IP state(s)
   CC_TRANS_PROP         true    necessary for Dyson orbitals job
   CC_DO_DYSON           true
   GUI                   2
$end

7.7.24 Interpretation of EOM/CI Wave Functions and Orbital Numbering

Analysis of the leading wave function amplitudes is always necessary for determining the character of the state (e.g., $\rm HOMO\rightarrow LUMO$ excitation, open-shell diradical, etc.). The CCMAN module print out leading EOM/CI amplitudes using its internal orbital numbering scheme, which is printed in the beginning. The typical CCMAN EOM-CCSD output looks like:

Root 1 Conv-d yes Tot Ene= -113.722767530 hartree (Ex Ene 7.9548 eV), 
U1^2=0.858795, U2^2=0.141205 ||Res||=4.4E-07
Right U1:
       Value                   i            ->    a
       0.5358                  7( B2  ) B   ->   17( B2  ) B
       0.5358                  7( B2  ) A   ->   17( B2  ) A
      -0.2278                  7( B2  ) B   ->   18( B2  ) B
      -0.2278                  7( B2  ) A   ->   18( B2  ) A

This means that this state is derived by excitation from occupied orbital #7 (which has $b_2$ symmetry) to virtual orbital #17 (which is also of $b_2$ symmetry). The two leading amplitudes correspond to $\beta \rightarrow \beta$ and $\alpha \rightarrow \alpha$ excitation (the spin part is denoted by $A$ or $B$ ). The orbital numbering for this job is defined by the following map:

The orbitals are ordered and numbered as follows:
Alpha orbitals:
Number  Energy    Type    Symmetry  ANLMAN number  Total number:
  0    -20.613     AOCC      A1      1A1     1
  1    -11.367     AOCC      A1      2A1     2
  2     -1.324     AOCC      A1      3A1     3
  3     -0.944     AOCC      A1      4A1     4
  4     -0.600     AOCC      A1      5A1     5
  5     -0.720     AOCC      B1      1B1     6
  6     -0.473     AOCC      B1      2B1     7
  7     -0.473     AOCC      B2      1B2     8

  0      0.071     AVIRT     A1      6A1     9
  1      0.100     AVIRT     A1      7A1     10
  2      0.290     AVIRT     A1      8A1     11
  3      0.327     AVIRT     A1      9A1     12
  4      0.367     AVIRT     A1     10A1     13
  5      0.454     AVIRT     A1     11A1     14
  6      0.808     AVIRT     A1     12A1     15
  7      1.196     AVIRT     A1     13A1     16
  8      1.295     AVIRT     A1     14A1     17
  9      1.562     AVIRT     A1     15A1     18
 10      2.003     AVIRT     A1     16A1     19
 11      0.100     AVIRT     B1      3B1     20
 12      0.319     AVIRT     B1      4B1     21
 13      0.395     AVIRT     B1      5B1     22
 14      0.881     AVIRT     B1      6B1     23
 15      1.291     AVIRT     B1      7B1     24
 16      1.550     AVIRT     B1      8B1     25
 17      0.040     AVIRT     B2      2B2     26
 18      0.137     AVIRT     B2      3B2     27
 19      0.330     AVIRT     B2      4B2     28
 20      0.853     AVIRT     B2      5B2     29
 21      1.491     AVIRT     B2      6B2     30

The first column is CCMAN’s internal numbering (e.g., 7 and 17 from the example above). This is followed by the orbital energy, orbital type (frozen, restricted, active, occupied, virtual), and orbital symmetry. Note that the orbitals are blocked by symmetries and then ordered by energy within each symmetry block, (i.e., first all occupied $a_1$ , then all $a_2$ , etc.), and numbered starting from 0. The occupied and virtual orbitals are numbered separately, and frozen orbitals are excluded from CCMAN numbering. The two last columns give numbering in terms of the final ANLMAN printout (starting from 1), e.g., our occupied orbital #7 will be numbered as 1 $B_2$ in the final printout. The last column gives the absolute orbital number (all occupied and all virtuals together, starting from 1), which is often used by external visualization routines.

CCMAN2 numbers orbitals by their energy within each irrep keeping the same numbering for occupied and virtual orbitals. This numbering is exactly the same as in the final printout of the SCF wave function analysis. Orbital energies are printed next to the respective amplitudes. For example, a typical CCMAN2 EOM-CCSD output will look like that:

 EOMEE-CCSD transition 2/A1
 Total energy = -75.87450159 a.u.  Excitation energy = 11.2971 eV.
 R1^2 = 0.9396  R2^2 = 0.0604  Res^2 = 9.51e-08

 Amplitude    Orbitals with energies
  0.6486       1 (B2) A                  ->    2 (B2) A                   
              -0.5101                          0.1729                     
  0.6486       1 (B2) B                  ->    2 (B2) B                   
              -0.5101                          0.1729                     
 -0.1268       3 (A1) A                  ->    4 (A1) A                   
              -0.5863                          0.0404                     
 -0.1268       3 (A1) B                  ->    4 (A1) B                   
              -0.5863                          0.0404

which means that for this state, the leading EOM amplitude corresponds to the transition from the first b $_2$ orbital (orbital energy $-0.5101$ ) to the second b $_2$ orbital (orbital energy 0.1729).

The most complete analysis of EOM-CC calculations is afforded by deploying a general wave-function analysis tool contained in the libwa module and described in Section 11.2.6. The EOM-CC state analysis is activated by setting STATE_ANALYSIS=TRUE. In addition, keywords controlling calculations of state and interstate properties should be set up accordingly.

Note: Wave function analysis is only available for CCMAN2.

Example 7.179 Wave function analysis of the EOM-IP states (He $_3^+$ ).

$molecule
   0 1
   He
   He   1       R1
   He   2       R1      1       A

   R1 = 1.236447
   A  = 180.00
$end

$rem
   METHOD            = EOM-CCSD
   BASIS             = 6-31G
   IP_STATES         = [1,0,0,0,0,1,0,0]
   CC_EOM_PROP       = true   Analyze state properties (state OPDM)
   CC_STATE_TO_OPT   = [1,1]  Compute transition properties wrt 1st EOM state of 1st irrep
   CC_TRANS_PROP     = true   Analyze transitions (transition OPDM)
   STATE_ANALYSIS    = true
   MOLDEN_FORMAT     = true
   NTO_PAIRS         = 2    
$end

0	Do not pre-converge
N	Perform N Davidson iterations pre-converging doubles.

1	Compute 2PA using the fastest algorithm (use $\tilde{\sigma }$ -intermediates for canonical
	and $\sigma$ -intermediates for RI/CD response calculations).
2	Use $\sigma$ -intermediates for 2PA response equation calculations.
3	Use $\tilde{\sigma }$ -intermediates for 2PA response equation calculations.

1	NYI for EOM-CC
2	Compute NACs using Szalay’s approach (this what needs to be specified for EOM-CC).

TRUE	Use the point group symmetry of the molecule
FALSE	Do not use point group symmetry (all states will be of $A$ symmetry).

EOM-CCSD(DT)	EOM-CCSD(dT), available for EE, SF, and IP
EOM-CCSD(FT)	EOM-CCSD(fT), available for EE, SF, IP, and EA
EOM-CCSD(ST)	EOM-CCSD(sT), available for IP

[spin, irrep, state]
spin = 0	Addresses states with low spin,
	see also EE_SINGLETS or IP_STATES,EA_STATES.
spin = 1	Addresses states with high spin,
	see also EE_TRIPLETS.
irrep	Specifies the irreducible representation to which
	the state belongs, for $C_{2v}$ point group symmetry
	irrep = 1 for $A_1$ , irrep = 2 for $A_2$ ,
	irrep = 3 for $B_1$ , irrep = 4 for $B_2$ .
state	Specifies the state number within the irreducible
	representation, state = 1 means the lowest excited
	state, state = 2 is the second excited state, etc..
0, 0, -1	Ground state.

TRUE	Find a point on the intersection seam and stop.
FALSE	Perform a minimization of the intersection seam.