7.10 Coupled-Cluster Excited-State and Open-Shell Methods 7.10.19 Non-Hartree-Fock Orbitals in EOM Calculations 7.10.21 EOM-CC Optimization and Properties Job Control

7.10.20 Analytic Gradients and State and Transition Properties for the CCSD and EOM-XX-CCSD Methods

(July 14, 2022)

The coupled-cluster package in Q-Chem can calculate properties of target EOM states including permanent dipoles, angular momentum projections, static polarizabilities, $\langle\hat{S}^{2}\rangle$ and $\langle\hat{R}^{2}\rangle$ values, g-tensors (CCSD only), nuclear gradients (and geometry optimizations). The target state of interest is selected by CC_STATE_TO_OPT in the $rem section, 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, electronic circular dichroism (ECD) rotatory strengths).

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 ⁷⁰⁷ Levchenko S. V., Wang T., Krylov A. I.
J. Chem. Phys.
(2005), 122, pp. 224106. Link and RI/Cholesky representation of the two-electron integrals (see also Section 6.15). 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 can calculate permanent and transition dipole moments, oscillator strengths, ECD rotatory strengths [both in length gauge (origin dependent) and in velocity gauge (origin independent)], $\langle\hat{R}^{2}\rangle$ (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 $\langle\hat{S}^{2}\rangle$ values. Calculation of g-tensors is available for CCSD wave functions (see Section 10.12.4 and example 10.12.4.4). 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, $\langle\hat{R}^{2}\rangle$ , $\langle\hat{S}^{2}\rangle$ 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 7.10.28 and 10.2.9. 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 input section allows the user to specify precisely which properties and for which pairs of states to computed. When the $trans_prop section is present in the input, it overrides the setting of the CC_TRANS_PROP $rem variable. However, for $trans_prop to work, CC_TRANS_PROP does need to be set.

$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
$end

Notes about the $trans_prop input section:

1.

calc computes properties for the first pair list (or state list) before it.
2.

The pair list is optional: if there is no pair list, all possible combinations within the state list will be considered.
3.

Options after calc include: nac, soc, dyson, 2pa, dipole, default, pcm, opdm_norm, wfa, ang, linmom, dipole linmom, ecd. Currently, only some of them are implemented.

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

7.10.20.1 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.
$end

7.10.20.2 EOM-CC Oscillator Strengths in Velocity and Mixed Gauges

The oscillator strength of one-photon absorption is usually calculated in the length gauge (“lg”)

f^{\rm{lg}}=\frac{2\omega_{fn}}{3}\sum_{\alpha=x,y,z}\langle n|\hat{\mu}_{% \alpha}|f\rangle\langle f|\hat{\mu}_{\alpha}|n\rangle,

(7.61)

where $\hat{\mu}$ is the electric dipole moment operator, but it can also be expressed in velocity gauge (“vg”) and mixed gauge (“mx”) as follows

f^{\rm{vg}}=\frac{2}{3\omega_{fn}}\sum_{\alpha=x,y,z}\langle n|\hat{p}_{\alpha% }|f\rangle\langle f|\hat{p}_{\alpha}|n\rangle,

(7.62)

f^{\rm{mx}}=\frac{2i}{3}\sum_{\alpha=x,y,z}\langle n|\hat{\mu}_{\alpha}|f% \rangle\langle f|\hat{p}_{\alpha}|n\rangle,

(7.63)

where $\hat{p}$ is the linear momentum operator.

All three gauges are implemented as part of the EOM-CCSD transition properties in Q-Chem. Whereas $f^{\rm lg}$ is computed as the default transition property, the other two must be activated. This is done in the $trans_prop input section by specifying calc linmom for velocity gauge only, or calc dipole linmom for all three gauges. This is illustrated in Example 7.10.21.1.

Note that:

–

If the calculation of ECD is activated in the input, $f^{\rm lg}$ , $f^{\rm vg}$ , and $f^{\rm mx}$ will also be computed;
–

If the energy separation between the two states is very small (threshold = $10^{-6}$ ), as in case of (near)-degenerate states, a warning is issued and the calculation of the velocity gauge strength is skipped.

7.10.20.3 Calculations of EOM-CC Rotatory Strengths (ECD and XCD) in Length and Velocity Gauges

Q-Chem affords ECD and XCD calculations using EOM-CC wavefunctions using formalism and implementation described in Ref. 40 Andersen J. A. et al.
J. Chem. Theory Comput.
(2022), 18, pp. 1748. Link . The rotatory strength of ECD within CC theory is defined as:

	$\displaystyle R^{\text{lg}}_{nf}$	$\displaystyle=-\frac{1}{4}\sum_{\alpha=x,y,z}\left\{\langle\hat{\mu}^{nf}_{% \alpha}\rangle\langle\hat{L}^{fn}_{\alpha}\rangle-\langle\hat{L}^{nf}_{\alpha}% \rangle\langle\hat{\mu}^{fn}_{\alpha}\rangle\right\},$		(7.64)
	$\displaystyle R^{\text{vg}}_{fn}$	$\displaystyle=-\frac{1}{4\omega_{nf}}\sum_{\alpha=x,y,z}\left\{\langle\hat{p}^% {nf}_{\alpha}\rangle\langle\hat{L}^{fn}_{\alpha}\rangle+\langle\hat{L}^{nf}_{% \alpha}\rangle\langle\hat{p}^{fn}_{\alpha}\rangle\right\}$		(7.65)

in the length (“lg”) and velocity (“vg”) formulations, respectively. $\hat{\mu}$ is the electric dipole moment operator and $\hat{p}$ is the linear momentum operator. The length-gauge expression in Eq. (7.64) is gauge-origin dependent, whereas Eq. (7.65) is gauge-origin independent.

ECD is available for ground-state and excited-state EOM-EE-CCSD transitions in the valence and core-excited regimes. The latter is based on the fc-CVS-EOM-EE-CCSD framework. The ECD calculations are not available for EOM-IP/EOM-DIP calculations yet.

The computation of rotatory strengths is activated in the input by either activating CC_EOM_ECD or via $trans_prop section, as illustrated in Example 7.10.21.1. In the latter case, the user can either request the calculation of the transition dipole moment, linear momentum, and angular momentum operators together (“calc dipole linmom ang”) or request ECD only (“calc ecd”).

Note that:

–

Rotatory strengths will be computed in both length and velocity gauge;
–

Oscillator strengths in length, velocity, and mixed gauges will be computed as well;
–

If the energy separation between the two states is very small (threshold = $10^{-6}$ ), as in the case of (near)-degenerate s states, a warning is issued and the calculation of the velocity gauge strength is skipped.

7.10.20.4 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. ³¹⁵ Epifanovsky E. et al.
J. Chem. Phys.
(2015), 143, pp. 064102. Link ^, ⁹⁶² Pokhilko P., Epifanovsky E., Krylov A. I.
J. Chem. Phys.
(2019), 151, pp. 034106. Link 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 molecule, ⁹⁶⁶ Pokhilko P. et al.
J. Phys. Chem. A
(2019), 123, pp. 482. Link the relative phases of these matrix elements are lost. The new version of the code solves this issue and also improves performance. ⁹⁶² Pokhilko P., Epifanovsky E., Krylov A. I.
J. Chem. Phys.
(2019), 151, pp. 034106. Link 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

	$\displaystyle T^{1,1}_{p,q}=-a^{\dagger}_{p\alpha}a_{q\beta}$		(7.66)
	$\displaystyle T^{1,0}_{p,q}=\frac{1}{\sqrt{2}}\left(a^{\dagger}_{p\alpha}a_{q% \alpha}-a^{\dagger}_{p\beta}a_{q\beta}\right)$		(7.67)
	$\displaystyle T^{1,-1}_{p,q}=a^{\dagger}_{p\beta}a_{q\alpha}$		(7.68)

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

\langle\Psi_{I}SM|T^{1,k}_{p,q}|\Psi_{J}S^{\prime}M^{\prime}\rangle=\langle S^% {\prime}M^{\prime}|1k|SM\rangle\langle\Psi_{I}S||T^{1,\cdot}_{p,q}||\Psi_{J}S^% {\prime}\rangle,

(7.69)

where $S$ and $S^{\prime}$ denote spins, and $M$ and $M^{\prime}$ denote spin projections of $\Psi_{I}$ and $\Psi_{J}$ , respectively. The reduced matrix elements form a new, spinless triplet transition density matrix, which we will denote as $u^{1,\cdot}$ . The spin-orbit reduced matrix elements are obtained as a trace with $u^{1,\cdot}$ :

	$\displaystyle\langle\Psi_{I}S\|\|H_{L_{-}}\|\|\Psi_{J}S^{\prime}\rangle=-\frac{1}{% 2}\sum_{p,q}h^{SO}_{L_{-};p,q}u^{1,\cdot}_{p,q}$		(7.70)
	$\displaystyle\langle\Psi_{I}S\|\|H_{L_{0}}\|\|\Psi_{J}S^{\prime}\rangle=\frac{% \sqrt{2}}{2}\sum_{p,q}h^{SO}_{L_{z};p,q}u^{1,\cdot}_{p,q}$		(7.71)
	$\displaystyle\langle\Psi_{I}S\|\|H_{L_{+}}\|\|\Psi_{J}S^{\prime}\rangle=+\frac{1}{% 2}\sum_{p,q}h^{SO}_{L_{+};p,q}u^{1,\cdot}_{p,q},$		(7.72)

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 7.7.84 Example of an output of the new SOC code for a transition between triplet and singlet states (open-shell reference). Only A $\to$ B 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, A $\to$ B and B $\to$ A transitions lead to, in general, different properties. In the end of the considered transition the matrix, averaged between A $\to$ B and B $\to$ A is printed with phases, corresponding to A $\to$ B 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>
 <Sz=0.00|(1.536309,12.805673)(0.000000,-11.558274)(1.536309,-12.805673)
 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>
 <Sz=0.00|(1.176458,5.735480)(0.000000,-6.173180)(1.176458,-5.735480)

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 $\hat{S}^{2}$ . 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. Examples 7.10.21.1 and 7.10.21.1 illustrate calculation of SOC using SOMF in the acetylene-O complex.

Note: In the case of ECP, attempts to compute SOCs using the above algorithms (with CALC_SOC=1 or 2) will, most likely, result in unphysically low SOC values. Instead, we recommend using effective charges (CALC_SOC=4).

To activate the NTO analysis of the spinless one-particle transition density matrices ⁹⁶⁴ Pokhilko P., Krylov A. I.
J. Phys. Chem. Lett.
(2019), 10, pp. 4857–4862. Link and to compute spin-orbit integrals over NTOs, set STATE_ANALYSIS = TRUE , MOLDEN_FORMAT = TRUE, in addition to CALC_SOC = 1, 2, or 4.

Q-Chem also affords computing SOCs using one-electron SO operator with effective nuclear charges ³³⁰ Fedorov D. G. et al.
Int. Rev. Phys. Chem.
(2003), 22, pp. 551. Link . The rational for this approach is that the effect of the mean-field two-electron part of the SOC operator can be approximated by screened nuclear charges. In the case of all-electron calculations (no ECP), this approach is less rigorous than SOMF, but it does not use two-electron integrals at all and is, therefore, less expensive. Moreover, using effective charges allows one to obtain reasonable values of SOCs with ECPs. This calculation is activated by using CALC_SOC=4. Q-Chem output will contain one-electron SOCs computed with the original and screened (effective) nuclear charges. In the case of open-shell references, SOCs are computed using $L_{+}/L_{-}$ averaging (as for the CALC_SOC=1 case). By default, the calculations use effective charges from Fedorov et al ³³⁰ Fedorov D. G. et al.
Int. Rev. Phys. Chem.
(2003), 22, pp. 551. Link (tabulated for lithium through astatine, except lanthanides): for all-electron calculations, the effective charges from the 6-31G basis are used, and for ECPs, SBKJC charges are used. Alternatively, the user can provide user-defined effective charges via input section $soc_eff_charges, as follows:

$soc_eff_charges
   8.0  6.0   !use eff. charge=6 for oxygen (Z=8)
   17.0 11.0 !use eff. charge=11 for chlorine (Z=17)
$end

Examples 7.10.21.1 and 7.10.21.1 illustrate this capability.

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.

Note: The most flexible control for computing transition properties is afforded by the $trans_prop input section, as described in Section 7.10.20.

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

Calculations of nonadiabatic (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”: ¹¹⁷² Tajti A., Szalay P. G.
J. Chem. Phys.
(2009), 131, pp. 124104. Link

h^{x}_{IJ}\equiv\langle\Psi_{I}|H^{x}|\Psi_{J}\rangle=\frac{1}{2}\left(G_{(I+J% )}-G_{I}-G_{J}\right),

(7.73)

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. ³²⁵ Faraji S., Matsika S., Krylov A. I.
J. Chem. Phys.
(2018), 148, pp. 044103. Link

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

7.10.20.6 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. 851 Nanda K. D., Krylov A. I.
J. Chem. Phys.
(2015), 142, pp. 064118. Link . This feature is available both for canonical and RI/CD implementations. The undamped canonical 2PA cross sections between EOM-DEA-CCSD states are also available. 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 $\delta^{TPA}$ (also known as rotationally averaged 2PA strength), as defined in Eq. (30) of Ref. 851 Nanda K. D., Krylov A. I.
J. Chem. Phys.
(2015), 142, pp. 064118. Link .

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): $2h\nu=E_{ex}$ 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 $\epsilon$ 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$^{-1}$
$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 ⁸⁵² Nanda K. D., Krylov A. I.
J. Phys. Chem. Lett.
(2017), 8, pp. 3256. Link . 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. 852 Nanda K. D., Krylov A. I.
J. Phys. Chem. Lett.
(2017), 8, pp. 3256. Link for more details).

Undamped 2PA response wave functions are computed using the (new) iterative Davidson-like subspace procedure (called Dalton solver) ²¹⁴ Christiansen O. et al.
J. Chem. Phys.
(1998), 108, pp. 2801. Link ^, ⁵⁸⁷ Kauczor J. et al.
J. Chem. Phys.
(2013), 139, pp. 211102. Link , which is controlled by the DALTON_XCONV, DALTON_PRECOND_START, DALTON_MAXITER, and DALTON_MAXSPACE keywords. Damped 2PA response wave functions are computed using the (old) iterative DIIS procedure, which is controlled by the CC_MAX_ITER, CC_DIIS_START, CC_DIIS_SIZE, CC_EOM_2PA_ECONV, and CC_EOM_2PA_XCONV.

2PA calculations can be carried out using single precision, by activating CC_EOM_2PA_SINGLE_PREC keyword, in addition to the keywords activating single-precision execution of CC and EOM-CC equations (see Section 7.10.13. Capabilities for computing other non-linear properties include RIXS (see Section 7.10.8.1) and static and dynamical polarizabilities (Section 7.10.20.8).

7.10.20.7 Calculation of Optical Rotation for CCSD Wave Function

Q-Chem affords calculations of the optical rotation (OR) tensor and the related specific optical rotation ⁹²⁰ Pecul M., Ruud K.
Adv. Quantum Chem.
(2005), 50, pp. 185. Link (at a chosen frequency corresponding to the polarimeter’s operating lamp) for a CCSD wave function (in CCMAN2) ⁴⁰ Andersen J. A. et al.
J. Chem. Theory Comput.
(2022), 18, pp. 1748. Link . The theory is based on the expectation-value approach, i.e., the sum-over-states expressions recast into closed-form expressions using response states ⁴⁰ Andersen J. A. et al.
J. Chem. Theory Comput.
(2022), 18, pp. 1748. Link . The calculation involves computing regular or damped response wave functions. Length gauge, velocity gauge, and modified velocity gauge are used ⁹²¹ Pedersen T. B. et al.
Chem. Phys. Lett.
(2004), 393, pp. 319. Link .

The calculation of the regular (i.e., undamped) optical rotation is deployed by setting CC_OPTROT to 5 and involves computing (undamped) density matrices using zero-order and response wave functions. If the frequency of probing linearly-polarized (lp) light is nearly resonant with an excitation energy, the undamped response calculations diverge. By augmenting the frequency with a phenomenological imaginary damping factor (e.g., 0.001 hartrees), the corresponding damped response wave functions are computed.

In the length (velocity) gauge, the real (imaginary) part of the damped optical rotation tensor corresponds to the differential absorption of the two circularly polarized components of the lp light, i.e., to the electronic circular dichroism (ECD). The imaginary (real) part corresponds to the dispersion (ORD). Currently, these features are available for the canonical implementation only.

Note: CC_FULLRESPONSE must be set to 0 and CC_REF_PROP must be set to TRUE for these calculations.

The response equations for OR are solved using the same solver as the response equations for polarizabilities (see Section 7.10.20.8 for relevant keywords).

A typical $opt_rot section input for a regular (undamped) optical rotation calculation at the sodium D-line (589.6 nm = 16960.651 cm ${}^{-1}$ ) is:

$opt_rot
omega_1  16960651 0 1    !One calculation at D-line
damped_epsilon 0.0       ! Disables damped OR [mandatory].
length_gauge <0 or 1>    ! Computes length-gauge OR if set to 1
velocity_gauge <0 or 1>  ! Computes (modified) velocity gauge OR if set to 1
$end

For a damped calculation, the frequency range, damping factor, and gauge choice are selected using the $opt_rot section as follows:

$opt_rot
omega_1  500000 10000 6  !Scans 500 cm$^{-1}$ to 550 cm$^{-1}$
                         !      in steps of 10 cm$^{-1}$
damped_epsilon 0.01      ! (Optional) Use for damped CCSD OR only.
                         ! Sets a damping factor of 0.01 hartrees.
length_gauge <0 or 1>    ! Computes length-gauge OR if set to 1
velocity_gauge <0 or 1>  !Computes (modified) velocity gauge OR if set to 1
$end

Note: Like other transition properties, ECD/XCD spectra can also be computed from the rotatory strengths between EOM-CC/CVS-EOM-CC states. See Section 7.10.20.

7.10.20.8 Calculations of Static and Dynamical Polarizabilities for CCSD and EOM-CCSD Wave Functions

Calculation of the static 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 energy ⁸⁵⁶ Nanda K., Krylov A. I.
J. Chem. Phys.
(2016), 145, pp. 204116. Link as well as using the expectation-value approach, i.e., the sum-over-states expression for the exact wave function ⁸⁵⁰ Nanda K. D., Krylov A. I., Gauss J.
J. Chem. Phys.
(2018), 149, pp. 141101. Link . 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. CCSD static and dynamical polarizabilities within the damped response theory are also available, which can be turned on by specifying the damped_epsilon variable in the $dyn_pol section.

$dyn_pol
   500000 10000 6  !Scans 500 cm$^{-1}$ to 550 cm$^{-1}$
                   !      in steps of 10 cm$^{-1}$
damped_epsilon 10  !(Optional) Use for damped CCSD polarizabilities only.
                   !Sets a damping factor of 10/1000 hartrees.
$end

Currently, these features are available for the canonical implementation only.

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

Undamped polarizability response wave functions are computed using the (new) iterative Davidson-like subspace procedure (called Dalton solver) ²¹⁴ Christiansen O. et al.
J. Chem. Phys.
(1998), 108, pp. 2801. Link ^, ⁵⁸⁷ Kauczor J. et al.
J. Chem. Phys.
(2013), 139, pp. 211102. Link , which is controlled by the DALTON_XCONV, DALTON_PRECOND_START, DALTON_MAXITER, and DALTON_MAXSPACE keywords. Damped polarizability response wave functions are computed using the (old) DIIS solver controlled by CC_MAX_ITER, CC_DIIS_START, CC_DIIS_SIZE, and CC_CONVERGENCE.

Note: Dalton solver uses square of the norm of the residual in estimating convergence (as in the Davidson solver), whereas the DIIS solver uses the norm of the difference of the response vector between iterations. The default threshold for convergence in response calculations (2PA, polarizability, RIXS) is 5.

7.10.20.9 Calculations of Static and Dynamical First Hyperpolarizabilities for CCSD Wave Functions

Calculation of the static and dynamical dipole first hyperpolarizabilities for CCSD wave functions is available in CCMAN2 (currently, canonical implementation only). First hyperpolarizabilities are computed within the damped-response expectation-value approach ⁸⁵⁴ Nanda K. D., Krylov A. I.
J. Chem. Phys.
(2021), 154, pp. 184109. Link . Setting CC_1HPOL to 1 deploys the algorithm that computes this property with the least number of response wave-function calculations. Setting CC_1HPOL to 3, computes this property by constructing the second-order response density matrices, which involves computing a large number of second-order response wave functions. These second-order response density matrices are then used to compute second-order response natural orbitals using libwfa for quantitative wave-function and orbital analyses. Calculations of first hyperpolarizabilities require the $1hpol section, which specifies the frequency ranges of the two photons ( $\omega_{1}$ and $\omega_{2}$ ; $\omega_{3}=\omega_{1}+\omega_{2}$ ), as well as the corresponding damping factors.