11.15 General Response Theory

Many of the preceding sections of chapter 11 are concerned with properties that require the solution of underlying equations similar to those from TDDFT (see eq. eq:TDDFT), but in the presence of a (time-dependent) perturbation:

$\begin{equation} \left[ \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^{*} & \mathbf{A}^{*} \end{pmatrix} - \omega _{f} \begin{pmatrix} \mathbf{\Sigma } & \mathbf{\Delta } \\ -\mathbf{\Delta }^{*} & -\mathbf{\Sigma }^{*} \end{pmatrix} \right] \begin{pmatrix} \mathbf{X} \\ \mathbf{Y} \end{pmatrix} = \begin{pmatrix} \mathbf{V} \\ -\mathbf{V}^{*} \end{pmatrix} , \end{equation}$

(11.87)

where $\mathbf{\Sigma }\rightarrow \mathbf{0}$ and $\mathbf{\Delta }\rightarrow \mathbf{1}$ for canonical HF/DFT MOs. The functionality for solving these equations with a general choice of operators representing a perturbation $\mathbf{V}$ is now available in Q-Chem. Both singlet[Jørgensen et al.(1988)Jørgensen, Jensen, and Olsen] and triplet[Olsen et al.(1989)Olsen, Yeager, and Jørgensen] response are available for a variety of operators (see table 11.4).

An additional feature of the general response module is its ability to work with non-orthogonal MOs. In a formulation analogous to TDDFT(MI)[Liu and Herbert(2015)], the linear response for molecular interactions[Berquist and Lambrecht()], or LR(MI), method is available to solve the linear response equations on top of ALMOs.

The response solver can be used with any density functional available in Q-Chem, including range-separated functionals (e.g. CAM-B3LYP, $\omega$ B97X) and meta-GGAs (e.g. M06-2X).

There are a few limitations:

No post-HF/correlated methods are available yet.
Currently, only linear response is implemented.
Only calculations on top of restricted and unrestricted (not restricted open-shell) references are implemented.
Density functionals including non-local dispersion (e.g. VV10, $\omega$ B97M-V) are not yet available.

11.15.1 Job Control

Only one keyword is necessary in the $rem section to activate the response module. All other options are controlled through the $response input section.

RESPONSE

Activate the general response property module.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

FALSE (or 0)

Don’t activate the general response property module.

TRUE (or 1)

Activate the general response property module.

RECOMMENDATION:

None.

ORDER

Sets the maximum order of response theory to perform.

INPUT SECTION: $response
TYPE:

STRING

DEFAULT:

LINEAR

OPTIONS:

LINEAR

Perform up through linear response.

RECOMMENDATION:

None. Currently, only linear response is implemented.

SOLVER

Sets the algorithm for solving the response equations.

INPUT SECTION: $response
TYPE:

STRING

DEFAULT:

DIIS

OPTIONS:

LINEAR

Iteratively solve the response equations without convergence acceleration.

DIIS

Iteratively solve the response equations using DIIS for convergence acceleration.

RECOMMENDATION:

DIIS

HAMILTONIAN

Sets the approximation used for the orbital Hessian.

INPUT SECTION: $response
TYPE:

STRING

DEFAULT:

RPA

OPTIONS:

RPA

No approximations.

TDA

Same as the CIS approximation.

CIS

Synonym for TDA.

RECOMMENDATION:

None.

SPIN

Does the operator access same spin (singlet) or different spin (triplet) states?

INPUT SECTION: $response
TYPE:

STRING

DEFAULT:

SINGLET

OPTIONS:

SINGLET

Operator is spin-conserving.

TRIPLET

Operator is not spin-conserving.

RECOMMENDATION:

None. Care must be taken as all operators in a single calculation will be forced to follow this option.

MAXITER

Maximum number of iterations.

INPUT SECTION: $response
TYPE:

INTEGER

DEFAULT:

60

OPTIONS:

$n$

Maximum number of iterations.

RECOMMENDATION:

Use the default value.

CONV

Convergence threshold. For the DIIS solver, this is the DIIS error norm. For the linear solver, this is the response vector RMSD between iterations.

INPUT SECTION: $response
TYPE:

INTEGER

DEFAULT:

8

OPTIONS:

$n$

Sets the convergence threshold to $10^{-n}$ .

RECOMMENDATION:

Use the default value.

DIIS_START

Iteration number to start DIIS. Before this, linear iterations are performed.

INPUT SECTION: $response
TYPE:

INTEGER

DEFAULT:

1

OPTIONS:

$n$

Iteration number to start DIIS.

RECOMMENDATION:

Use the default value.

DIIS_VECTORS

Maximum number of DIIS vectors to keep.

INPUT SECTION: $response
TYPE:

INTEGER

DEFAULT:

7

OPTIONS:

$n > 0$

Maximum number of DIIS vectors to keep.

RECOMMENDATION:

Use the default value.

RHF_AS_UHF

Should the response equations be solved as though an unrestricted reference is being used?

INPUT SECTION: $response
TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE

Treat an RHF wavefunction as though it were UHF.

FALSE

Treat an RHF wavefunction as RHF.

RECOMMENDATION:

Use the default value. Only useful for debugging.

PRINT_LEVEL

Sets a general printing level across the response module.

INPUT SECTION: $response
TYPE:

INTEGER

DEFAULT:

2

OPTIONS:

1

Print the initial guess and the final results.

2

1 + iterations and comments.

10

Kill trees.

RECOMMENDATION:

Use the default value.

RUN_TYPE

Should a single response calculation be performed, or should all permutations of the orbital Hessian and excitation type be performed?

INPUT SECTION: $response
TYPE:

STRING

DEFAULT:

SINGLE

OPTIONS:

SINGLE

Use only the orbital Hessian and excitation type specified in their respective keywords.

ALL

Use all permutations of RPA/TDA and singlet/triplet.

RECOMMENDATION:

Use the default value, unless a comparison between approximations and excitation types is desired.

SAVE

Save any quantities to disk?

INPUT SECTION: $response
TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0

Don’t save any quantities to disk.

1

Save quantities in MO basis.

2

Save quantities in MO and AO bases.

RECOMMENDATION:

None.

READ

Read any quantities from disk?

INPUT SECTION: $response
TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0

Don’t read any quantities from disk.

1

Read quantities in MO basis.

2

Read quantities in AO basis.

RECOMMENDATION:

None.

DUMP_AO_INTEGRALS

Should AO-basis property integrals be saved to disk?

INPUT SECTION: $response
TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE

Save AO-basis property integrals to disk.

FALSE

Don’t save AO-basis property integrals to disk.

RECOMMENDATION:

None.

FORCE_NOT_NONORTHOGONAL

Should the canonical response equations be solved, ignoring the identity of the underlying orbitals?

INPUT SECTION: $response
TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE

FALSE

RECOMMENDATION:

Leave as false. Using the standard (canonical) response equations with non-orthogonal MOs will give incorrect results.

FORCE_NONORTHOGONAL

Should the non-orthogonal response equations be solved, ignoring the identity of the underlying orbitals?

INPUT SECTION: $response
TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE

FALSE

RECOMMENDATION:

Leave as false. When used with canonical MOs, this should give the same answer as with the standard equations, but at greater computational cost.

FREQUENCY

Strength of one or more incident fields in atomic units. A separate response calculation will be performed for every field strength. 0.0 corresponds to the static limit.

INPUT SECTION: $response
TYPE:

DOUBLE

DEFAULT:

0.0

OPTIONS:

$l \, m \, n \, \dots$

One or more field strengths separated by spaces.

RECOMMENDATION:

None.

11.15.2 $response Section and Operator Specification

The specification of operators used in solving for response vectors is designed to be very flexible. The general form of the $response input section is given by

$response
  keyword_1 setting_1
  keyword_2 setting_2
  ...
  [operator_1_label, operator_1_origin]
  [operator_2_label, operator_2_origin]
  [operator_3_label, operator_3_origin]
  ...
$end

where the keywords are those found in section 11.15.1 (with the exception of RESPONSE).

The specification of an operator is given within a line contained by [], where the first element is a label from table 11.4, and the second element is a label from table 11.5. Operator specifications may appear in any order. Response values are calculated for all possible permutations of operators and their components.

For the Cartesian moment operator, a third field within [] may be specified for the order of the expansion, entered as $(i, j, k)$ . For example, the molecular response to the moment of order (2, 5, 4) with its origin at (0.2, 0.3, 0.4) a.u. can be found with the operator specification

[multipole, (0.2, 0.3, 0.4), (2, 5, 4)]

Table 11.4: Available operators

Operator Label	Description	Integral
`dipole` or `diplen`	dipole (length gauge)	$\ensuremath{\langle }\chi _{\mu } \| \mathbf{r}_{O} \| \chi _{\nu } \ensuremath{\rangle }$
`quadrupole`	second moment (length gauge)	$\ensuremath{\langle }\chi _{\mu } \| \mathbf{r} \mathbf{r}^{T} \| \chi _{\nu } \ensuremath{\rangle }$
`multipole`	arbitrary-order Cartesian moment (length gauge)	$\ensuremath{\langle }\chi _{\mu } \| x^{i} y^{j} z^{k} \| \chi _{\nu } \ensuremath{\rangle }$
`fermi` or `fc`	Fermi contact	$\frac{4\pi g_{e}}{3} \ensuremath{\langle }\chi _{\mu } \| \delta (\mathbf{r}_{K}) \| \chi _{\nu } \ensuremath{\rangle }$
`spindip` or `sd`	spin dipole	$\frac{g_{e}}{2} \ensuremath{\langle }\chi _{\mu } \| \frac{3\mathbf{r}_{K}\mathbf{r}_{K}^{T} - r_{K}^{2}}{r_{K}^{5}} \| \chi _{\nu } \ensuremath{\rangle }$
`angmom` or `dipmag`	angular momentum	$\ensuremath{\langle }\chi _{\mu } \| \mathbf{L}_{O} \| \chi _{\nu } \ensuremath{\rangle }$
`dipvel`	dipole (velocity gauge)	$\ensuremath{\langle }\chi _{\mu } \| \mathbf{\nabla } \| \chi _{\nu } \ensuremath{\rangle }$

Table 11.5: Available operator origins

Origin Label	Description
`zero`	Cartesian origin, same as (0.0, 0.0, 0.0)
`(x, y, z)`	arbitrary point (double precision, units are bohrs)

11.15.3 Examples Including $response Section

Example 11.270 Input for calculating all components of the static (dipole) polarizability at the Cartesian origin for tryptophan. All of the options given are defaults.

$molecule
0 1
N     -0.0699826875    0.3321987191    0.2821283177
C      1.3728035449    0.0970713322   -0.0129587739
C      2.0969275417   -0.0523593054    1.3682652221
O      3.1382490088   -0.6563684788    1.5380162924
C      1.9529664597    1.3136139853   -0.7956021969
H      1.8442727348    2.2050605044   -0.1801631789
H      1.3455899915    1.4594935008   -1.6885689523
C      3.4053646872    1.1270611844   -1.1918075237
C      4.4845249667    1.6235038050   -0.5598918002
N      5.6509089647    1.2379326369   -1.2284610654
H      6.6009314349    1.4112351003   -0.9028629397
C      5.2921619642    0.4356274269   -2.3131617003
C      3.8942019475    0.3557998019   -2.3263315791
C      3.2659168792   -0.3832607567   -3.3431309548
H      2.1864306677   -0.4577058843   -3.3815918670
C      4.0381762333   -1.0087512639   -4.2870993776
H      3.5696890585   -1.5824763141   -5.0755609734
C      5.4445159165   -0.9194874753   -4.2519002882
H      6.0229926396   -1.4277973542   -5.0130007062
C      6.0869576238   -0.2024044961   -3.2767702726
H      7.1656650647   -0.1287762497   -3.2458650647
H      4.5457621618    2.2425310766    0.3253979653
H     -0.5159777859    0.7478905868   -0.5487661007
H      1.5420526570   -0.8143939718   -0.5935463196
H     -0.5302278747   -0.5823989653    0.4084507634
O      1.4575846656    0.5996887308    2.4093500287
H      0.5990015339    0.8842421241    2.0047830456
$end

$rem
   METHOD           = hf
   BASIS            = sto-3g
   SCF_CONVERGENCE  = 9
   THRESH           = 12
   RESPONSE         = true
$end

$response
   ORDER             linear
   SOLVER            diis
   HAMILTONIAN       rpa
   SPIN              singlet
   MAXITER           60
   CONV              8
   DIIS_START        1
   DIIS_VECTORS      7
   RHF_AS_UHF        false
   PRINT_LEVEL       2
   RUN_TYPE          single
   FREQUENCY         0.0
   [dipole, zero]
$end

Example 11.271 Functionally identical input for calculating all components of the static (dipole) polarizability at the Cartesian origin for tryptophan.

$rem
 jobtype         =  polarizability
 method          =  hf
 basis           =  sto-3g
 scf_convergence =  9
 thresh          =  12
$end

$molecule
0 1
N     -0.0699826875    0.3321987191    0.2821283177
C      1.3728035449    0.0970713322   -0.0129587739
C      2.0969275417   -0.0523593054    1.3682652221
O      3.1382490088   -0.6563684788    1.5380162924
C      1.9529664597    1.3136139853   -0.7956021969
H      1.8442727348    2.2050605044   -0.1801631789
H      1.3455899915    1.4594935008   -1.6885689523
C      3.4053646872    1.1270611844   -1.1918075237
C      4.4845249667    1.6235038050   -0.5598918002
N      5.6509089647    1.2379326369   -1.2284610654
H      6.6009314349    1.4112351003   -0.9028629397
C      5.2921619642    0.4356274269   -2.3131617003
C      3.8942019475    0.3557998019   -2.3263315791
C      3.2659168792   -0.3832607567   -3.3431309548
H      2.1864306677   -0.4577058843   -3.3815918670
C      4.0381762333   -1.0087512639   -4.2870993776
H      3.5696890585   -1.5824763141   -5.0755609734
C      5.4445159165   -0.9194874753   -4.2519002882
H      6.0229926396   -1.4277973542   -5.0130007062
C      6.0869576238   -0.2024044961   -3.2767702726
H      7.1656650647   -0.1287762497   -3.2458650647
H      4.5457621618    2.2425310766    0.3253979653
H     -0.5159777859    0.7478905868   -0.5487661007
H      1.5420526570   -0.8143939718   -0.5935463196
H     -0.5302278747   -0.5823989653    0.4084507634
O      1.4575846656    0.5996887308    2.4093500287
H      0.5990015339    0.8842421241    2.0047830456
$end

FALSE (or 0)	Don’t activate the general response property module.
TRUE (or 1)	Activate the general response property module.

LINEAR	Iteratively solve the response equations without convergence acceleration.
DIIS	Iteratively solve the response equations using DIIS for convergence acceleration.

RPA	No approximations.
TDA	Same as the CIS approximation.
CIS	Synonym for TDA.

SINGLET	Operator is spin-conserving.
TRIPLET	Operator is not spin-conserving.

TRUE	Treat an RHF wavefunction as though it were UHF.
FALSE	Treat an RHF wavefunction as RHF.

1	Print the initial guess and the final results.
2	1 + iterations and comments.
10	Kill trees.

SINGLE	Use only the orbital Hessian and excitation type specified in their respective keywords.
ALL	Use all permutations of RPA/TDA and singlet/triplet.

0	Don’t save any quantities to disk.
1	Save quantities in MO basis.
2	Save quantities in MO and AO bases.

0	Don’t read any quantities from disk.
1	Read quantities in MO basis.
2	Read quantities in AO basis.

TRUE	Save AO-basis property integrals to disk.
FALSE	Don’t save AO-basis property integrals to disk.