7.10.20 Analytic Gradients and Properties for CCSD and EOM-XX-CCSD

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

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

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

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

All three gauges are implemented as part of the EOM-CCSD transition properties in Q-Chem. Whereas $f^{\lg }$ 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, $f^{\lg }$, $f^{\mathrm{vg}}$, and $f^{\mathrm{mx}}$ 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.

Q-Chem affords ECD and XCD calculations using EOM-CC wavefunctions using formalism and implementation described in Ref.  44 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:

in the length (“lg”) and velocity (“vg”) formulations, respectively. $\widehat{\mu }$ is the electric dipole moment operator and $\widehat{p}$ is the linear momentum operator. The length-gauge expression in Eq. (7.91) is gauge-origin dependent, whereas Eq. (7.92) 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.

Calculations of spin-orbit couplings (SOCs) for EOM-CC wave functions is available in CCMAN2. ³⁴¹ Epifanovsky E. et al.
J. Chem. Phys.
(2015), 143, pp. 064102. Link ^{, 1028} 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, ¹⁰³² 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. ¹⁰²⁸ 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

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

where $S$ and $S^{\prime }$ denote spins, and $M$ and $M^{\prime }$ denote spin projections of ${\mathrm{\Psi }}_I$ and ${\mathrm{\Psi }}_J$, 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 $u^{1,\cdot }$:

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.

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 ${\widehat{S}}^2$. 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  ¹⁰³⁰ 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 ³⁵⁶ 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 ³⁵⁶ 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.

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”: ¹²⁴⁸ Tajti A., Szalay P. G.
J. Chem. Phys.
(2009), 131, pp. 124104. Link

where, $G_I$, $G_J$, and $G_{IJ}$ are analytic gradients for states $I$, $J$, and a fictitious summed state $|{\mathrm{\Psi }}_{I+J}⟩\equiv |{\mathrm{\Psi }}_I⟩+|{\mathrm{\Psi }}_J⟩$. Currently, NACs for EE/IP/EA are available. ³⁵¹ 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.

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.  914 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.  914 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 $ϵ$ 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$^{-1}$
$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 ⁹¹⁵ 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.  915 Nanda K. D., Krylov A. I.
J. Phys. Chem. Lett.
(2017), 8, pp. 3256. Link for more details).

By default, both undamped and damped 2PA response wave functions are computed using a new iterative Davidson-like subspace procedure called the Dalton solver. ²³¹ Christiansen O. et al.
J. Chem. Phys.
(1998), 108, pp. 2801. Link ^{, 634} Kauczor J. et al.
J. Chem. Phys.
(2013), 139, pp. 211102. Link It is controlled by the DALTON_XCONV, DALTON_PRECOND_START, DALTON_MAXITER, and DALTON_MAXSPACE keywords. Damped 2PA response wave functions can also be computed using the older DIIS procedure by setting DAMPED_DALTON_SOLVER = FALSE. The DIIS solver is controlled by the CC_MAX_ITER, CC_DIIS_START, CC_DIIS_SIZE, CC_EOM_2PA_ECONV, and CC_EOM_2PA_XCONV.

Note:  The 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.

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 7.10.14. Capabilities for computing other non-linear properties include RIXS (see Section 7.10.8.1) and static and dynamic polarizabilities (Section 7.10.20.8).

Q-Chem affords calculations of the optical rotation (OR) tensor and the related specific optical rotation ⁹⁸⁴ 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) ⁴⁴ 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 ⁴⁴ 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 ⁹⁸⁵ 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.

Calculation of the static and dynamic 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 ⁹¹⁹ 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 ⁹¹³ 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.

Dynamic 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 dynamic 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.

By default, both undamped and damped polarizability response wave functions are computed using a new iterative Davidson-like subspace procedure called the Dalton solver. ²³¹ Christiansen O. et al.
J. Chem. Phys.
(1998), 108, pp. 2801. Link ^{, 634} Kauczor J. et al.
J. Chem. Phys.
(2013), 139, pp. 211102. Link It is controlled by the DALTON_XCONV, DALTON_PRECOND_START, DALTON_MAXITER, and DALTON_MAXSPACE keywords. Damped polarizability response wave functions can also be computed using the older DIIS solver by setting DAMPED_DALTON_SOLVER = FALSE. The DIIS solver is controlled by CC_MAX_ITER, CC_DIIS_START, CC_DIIS_SIZE, and CC _CONVERGENCE.

Note:  The 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.

Calculation of the static and dynamic 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 ⁹¹⁷ 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.

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

$1hpol
omega_1 0000000 1500000 3 0.01 !Scans omega_1 from 0 cm$^{-1}$ to (3-1)*1500 cm$^{-1}
                               ! in steps of 1500 cm$^{-1}
                               ! damping factor epsilon_1 is 0.01 hartrees
omega_2 0000000 2500000 4 0.01 !Scans omega_2 from 0 cm$^{-1}$ to (4-1)*2500 cm$^{-1}
                               ! in steps of 2500 cm$^{-1}
                               ! damping factor epsilon_2 is 0.01 hartrees
omega_3 0.01                   ! damping factor epsilon_3 is 0.01 hartrees
$end

Example 7.10.21.1 computes static and dynamic first hyperpolarizability for CCSD/STO-3G wave function for LiH using the framework of damped response theory and the expectation-value approach. The property for the following set of 3 photons is calculated (frequencies in cm${}^{-1}$): (0, 0; 0), (2500, 0; -2500), (5000, 0; -5000), (0, 3500; -3500), (0, 7000; -7000), (2500, 3500; -6000), (2500, 7000; -9500), (5000, 3500; -8500), (5000, 7000; -12000).

Q-Chem can compute quantities describing photo-ionization/photodetachment following the theoretical framework based on Dyson orbitals ⁴⁴⁴ Gozem S., Krylov A. I.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2022), 12, pp. e1546. Link . One can use computed Dyson orbitals and then calculate cross sections using the stand-alone ezDyson code ⁴⁴⁴ Gozem S., Krylov A. I.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2022), 12, pp. e1546. Link or request such calculations to be done by Q-Chem; the latter version is more efficient. These calculations require Dyson orbitals, which connect the initial and target states, and a wave-function of the free electron. Currently, only plane-wave treatment is available. This feature is only available in ccman2.

Total and angular-resolved photoionization cross sections can be computed by using the following set-up. First, the user should request calculation of appropriate initial and target states and the corresponding Dyson orbitals (using CC_DO_DYSON keyword) or the corresponding setting in $trans_prop section). Second, CC_FREE_ELECTRON should be set to TRUE. Third, the parameters for the cross sections—such as polarization of the light, energy range of photoelectrons, and details of orientational averaging—should be specified in the $free_electron section.

$free_electron
N_ENERGY_POINTS 20  !number of points in the kinetic energy grid
KE_MIN 10   !minimum kinetic energy of the free-electron in 1E-2  eV (here, 0.1 eV)
KE_MAX 1010  !maximum kinetic energy of the free-electron in 1E-2  eV  (here, 10.1 eV)
SPIN_DEG 2  !spin degeneracy factor
ORB_DEG 1  !orbital degeneracy factor
POL_X 0  !X-component of the polarization vector in 1E-2 (here, 0)
POL_Y 0  !Y-component of the polarization vector in 1E-2 (here, 0)
POL_Z 100  !Z-component of the polarization vector in 1E-2 (here, 1)
CIRC_POL 0  !if circular polarization is requested
DO_AVERAGE 0  !if orientational averaging to be performed
N_THETA 50  !number of theta-points in the angular grid
N_PHI 10  !number of phi-points in the angular grid
$end

Photoelectron angular distribution (PAD) — $\mathrm{\Omega }(\theta ,\varphi )$, where $\theta$ and $\varphi$ are angular coordinates of the photoelectron’s vector $𝐤$ in the lab frame — will be written in separate files for each ionized state (i.e., pad_1A1, pad_1B2, etc). For a linear polarization, asymmetry parameter $\beta$ will be calculated along with the cross section. Input parameters for the calculation are specified through the $free_electron section. The default values are: SPIN_DEG=2, ORB_DEG=1, and DO_AVERAGE=0 (no averaging). The calculations require a number of $\theta$ and $\varphi$ points to be specified by the user (defaults are 50 and 10 respectively) to determine the directions of the wave-vector $𝐤$ for computing PAD. The user should specify either the three components of the polarization (for linear polarization) or the circular polarization; the two options are mutually exclusive. For circular-polarized light, $z$ is the direction of light propargation. If DO_AVERAGE is set to 1, averaging over all molecular orinations is performed and polarization of light is assumed to be along the $z$-axis (the polarization setting will be ignored). These features are illustrated in examples 7.10.21.1-7.10.21.1 below.