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

7.9.18 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, angular momentum projections, static polarizabilities, S2 and R2 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, angular momentum matrix elements, 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 functionalityLevchenko:2005 and RI/Cholesky representation of the two-electron integrals (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.

Note:  Gradients for EOM-DIP/DEA-CCSD are not yet available.

For the CCSD and EOM-CCSD wave functions, Q-Chem currently can calculate permanent and transition dipole moments, oscillator strengths, R2 (as well as XX, YY and ZZ components separately, which is useful for assigning different Rydberg states, e.g., 3px vs. 3s, etc.), and the S2 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, angular momentum projections, R2, S2 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 7.9.26 and 10.2.6. 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 C1 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.

   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
   calc dipole soc    ! Compute transition dipole and SOC
   calc opdm_norm     ! Compute norm of transition OPDM

Notes about $trans_prop rem section:

  1. 1.

    calc computes properties for the first pair list (or state list) before it.

  2. 2.

    The pair list is optional: if there is no pair list, all possible combinations within the state list will be considered.

  3. 3.

    Options after calc include: nac, soc, dyson, 2pa, dipole, default, pcm, opdm_norm, wfa, ang. Currently, only some of them are 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. 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 quantity printed out is the microscopic cross-section δTPA (also known as rotationally averaged 2PA strength), as defined in Eq. (30) of Ref. Nanda:2015.

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): 2hν=Eex For resonant or near-resonant calculations (i.e., when one of the photons is in resonance with an excited state), damped response theory should be used; such calculations are deployed by adding DAMPED_EPSILON in the $2pa section (the value of ϵ is specified in hartrees, 0.001 is usually sufficient).

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

N_2PA_POINTS is the number of frequency pairs across the spectrum. The first value associated with OMEGA_1 is the frequency ×1000 in cm-1 at the start of the spectrum and the second value is the step size ×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: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 ωDMs (see Ref. Nanda:2017 for more details).

Capabilities for computing other non-linear properties include RIXS (see section and static and dynamic polarizabilities (section Calculations of Spin-Orbit Couplings Using EOM-CC Wave Functions

Calculations of spin–orbit couplings (SOCs) for EOM-CC wave functions is available in CCMAN2Epifanovsky:2015, Pokhilko:2019b. 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. There are two versions of the code that can be engaged by using CALC_SOC. The old code computes SOC as a matrix element between a pair of states that need to be explicitly computed; thus, it is able to produce only one matrix element per transition. Although this approach can be used to extract the whole matrix through rotations of the moleculePokhilko:2019a, the relative phases of these matrix elements are lost. The new version of the code solves this issue and also improves performancePokhilko:2019b. The new code actively uses Wigner–Eckart’s theorem to compute the entire spin–orbit matrix. We partition excitation operators by their spin properties. Since the spin–orbit operator is a triplet operator, we can consider the following triplet excitation operators

Tp,q1,1=-apαaqβ (7.52)
Tp,q1,0=12(apαaqα-apβaqβ) (7.53)
Tp,q1,-1=apβaqα (7.54)

If we apply Wigner–Eckart’s theorem, the following matrix elements are decomposed into the Clebsch–Gordan coefficients and reduced matrix elements:

ΨISM|Tp,q1,k|ΨJSM=SM|1k|SMΨIS||Tp,q1,||ΨJS, (7.55)

where S and S denote spins, and M and M denote spin projections of ΨI and ΨJ, respectively. The reduced matrix elements form a new, spinless triplet transition density matrix, which we will denote as u1,. The spin–orbit reduced matrix elements are obtained as a trace with u1,:

ΨIS||HL-||ΨJS=-12p,qhL-;p,qSOup,q1, (7.56)
ΨIS||HL0||ΨJS=22p,qhLz;p,qSOup,q1, (7.57)
ΨIS||HL+||ΨJS=+12p,qhL+;p,qSOup,q1,, (7.58)

Applying Wigner–Eckart’s theorem again, all possible matrix elements are generated from reduced matrix elements. This algorithm is available for the EOM-EE/SF/IP/EA wave functions for MP2 and CCSD references, as well as between the CCSD reference and EOM-EE/SF.

Example 7.68  Example of an output of the new SOC code for a transition between triplet and singlet states (open-shell reference). Only AB part is shown. First, one-electron reduced matrix elements and the actual matrix elements are printed. Note the proper symmetry of these elements. Second, mean-field reduced matrix elements are printed. Note a small violation of symmetry, which is restored for the averaged (symmetrized) reduced matrix elements. The actual matrix elements are printed in the end (in this version the symmetrized reduced matrix elements were used for its construction). Since EOM-CC methods are non-Hermitian, AB and BA transitions lead to, in general, different properties. In the end of the considered transition the matrix, averaged between AB and BA is printed with phases, corresponding to AB transition. To form a Hermitian Hamiltonian matrix, the user should take the averaged matrix, plug it into a corresponding block of a Hamiltonian matrix; the missing blocks are recovered from other transitions and complex conjugation.

   A(Ket)->B(Bra) transition SO matrices
 Analyzing Sz ans S^2 of the pair of states...
 Ket state:  Computed S^2 = 2.019877 will be treated as 2.000000 Sz = 0.000000
 Bra state:  Computed S^2 = 0.019216 will be treated as 0.000000 Sz = 0.000000
 Clebsch-Gordan coefficient: <1.000,0.000;1.000,0.000|0.000,0.000> = 0.577
 One-electron SO (cm-1)
 Reduced matrix elements:
 <S|| Hso(L-) ||S’> = (-2.660965,-22.180075)
 <S|| Hso(L0) ||S’> = (0.000000,-20.019517)
 <S|| Hso(L+) ||S’> = (-2.660965,22.180075)
 SOCC = 21.593626
 Actual matrix elements:
       |Sz=-1.00>           |Sz=0.00>           |Sz=1.00>
 Mean-field SO (cm-1)
 Reduced matrix elements:
 <S|| Hso(L-) ||S’> = (-2.069801,-9.910572)
 <S|| Hso(L0) ||S’> = (0.000000,-10.692261)
 <S|| Hso(L+) ||S’> = (-2.005570,9.957712)
 Singlet part of <S|| Hso(L0) ||S’> = (-0.000000,0.037101) (excluded from all matrix elements)
 L-/L+ Averaged reduced matrix elements:
 <S|| Hso(L-) ||S’> = (-2.037685,-9.934142)
 <S|| Hso(L+) ||S’> = (-2.037685,9.934142)
 SOCC = 10.328006
 Actual matrix elements:
       |Sz=-1.00>           |Sz=0.00>           |Sz=1.00>

The algorithm is activated by setting CALC_SOC to 1 or 2. The execution relies on a non-zero Clebsch–Gordan coefficient (otherwise, it is not possible to extract the necessary information from the transition density matrix). If the Clebsch–Gordan coefficient is zero, a warning will be printed, and the transition will be skipped. Usually it is possible to select the states, leading to non-zero Clebsch–Gordan coefficients, by selecting SF or EE, alpha or beta electrons in case of EA or IP. Determination of state multiplicities is based on S2. In case of a strong spin contamination, a warning is printed, and the closest multiplicity is assumed.

In the case of open-shell references, the resulting reduced matrix elements may violate symmetries implied by the spin–orbit operator. Setting CALC_SOC to 1 will produce these reduced matrix elements, but the actual spin–orbit matrix will be produced from the averaged reduced matrix elements, in which the desired symmetry is restored. Setting CALC_SOC to 2 will form spin–orbit matrix without averaging of reduced matrix elements. Although in practice the difference between these two options is small, the averaging may simplify the analysis of splittings.

Note:  The ECP version is not yet implemented. Attempts of using ECP will, most likely, result in unphysically low SOC values. A warning will be printed if an ECP is used.

To activate the NTO analysis of the spinless one-particle transition density matrices Pokhilko:2019c and to compute spin–orbit integrals over NTOs, set STATE_ANALYSIS = TRUE , MOLDEN_FORMAT = TRUE, in addition to CALC_SOC = 1 or 2.

The legacy code has the full two-electron treatment and the mean-field approximation. To enable the SOC calculation, transition properties between EOM states must be enabled via CC_TRANS_PROP, and SOC requested setting CALC_SOC to 3. 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. 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”:Szalay:2009

hIJxΨI|Hx|ΨJ=12(G(I+J)-GI-GJ), (7.59)

where, GI, GJ, and GIJ are analytic gradients for states I, J, and a fictitious summed state |ΨI+J|ΨI+|ΨJ. Currently, NACs for EE/IP/EA are available.Faraji:2018

Note:  Note that the individual components of the NAC vector depend on the molecular orientation. Calculations of Static and Dynamic Polarizabilities for CCSD and EOM-CCSD Wave Functions

Calculation of the and dynamical dipole polarizabilities for CCSD and EOM-EE/SF wave functions is available in CCMAN2. Polarizabilities are calculated as second derivatives of the CCSD or EOM-CCSD energyNanda:2016 as well as using the expectation-value approach, i.e., the sum-over-states expression for the exact wave functionNanda:2018. In the analytic derivative approach (or the response-theory formalism), only the response of the cluster amplitudes is taken into the account; orbital relaxation is not included. CC_FULLRESPONSE must be set to 2 to turn off the orbital relaxation but allow amplitude response in EOM calculations, if the polarizabilities are to be computed with this approach (both CC_POL and EOM_POL must equal 1 or 2). CC_POL and EOM_POL = 3 or 4 coupled with CC_FULLRESPONSE = 0 will enable expectation-value polarizabilities. Note that CC_REF_PROP and CC_EOM_PROP/ must be set to TRUE for these calculations.

Dynamical polarizabilities are enabled using the $dyn_pol section, which specifies the frequency range. Without this section, only the static polarizabilities will be computed.

   500000 10000 6  !Scans 500 cm$^{-1}$ to 550 cm$^{-1}$
                   !      in steps of 10 cm$^{-1}$

Currently, this feature is available for the canonical implementation only.

Note:  EOM-CCSD polarizabilities are available for the EE and SF wave functions only. Calculations of Angular Momentum Matrix Elements Using EOM-CC Wave Functions

Angular momentum matrix elements can be useful for analyzing wave functions, assigning state characters, or identifying term symmetry labels in linear molecules. As other one-electron properties, these matrix elements are evaluated using one-particle state and transition density matrices. Angular momentum is a gauge-dependent property, meaning that its value depends on the origin of the gauge. The gauge position can be controlled with $gauge_origin input section. The default position of the gauge corresponds to the (0,0,0) of the coordinate system.

$gauge_origin            Position of the gauge origin for
1.00  0.200  0.500       angular momentum calculations in angstroms.