*Code author: A. Eugene DePrince*

*Section author: A. Eugene DePrince*

*Module:* *Keywords*, *PSI Variables*, FNOCC

The computational cost of the QCISD(T), CCSD(T), CEPA, and MP4 methods can be reduced by constructing a compact representation of the virtual space based on the natural orbitals of second-order perturbation theory. The most demanding steps in the CCSD and (T) algorithms scale as and , where and represent the number of oribitals that are occupied and unoccupied (virtual) in the reference function, respectively. By reducing the the size of the virtual space, the cost of evaluating these terms reduces by a factor of , where represents the number of virtual orbitals retained after the FNO truncation.

The general outline for the FNO procedure in PSI4 is:

- construct the virtual-virtual block of the unrelaxed MP2 one-particle density matrix (OPDM)
- diagonalize this block of the OPDM to obtain a set of natural virtual orbitals
- based on some occupancy threshold, determine which orbitals are unimportant and may be discarded
- project the virtual-virtual block of the Fock matrix onto the truncated space
- construct semicanonical orbitals by diagonalizing the virtual-virtual block of the Fock matrix
- proceed with the QCISD(T) / CCSD(T) / MP4 computation in the reduced virtual space

A second-order correction based upon the MP2 energies in the full and
truncated spaces captures much of the missing correlation effects. More
details on the implementation and numerical accuracy of FNO methods in
PSI4 can be found in Ref. [DePrince:2013:293]. FNO computations
are controlled through the keywords *NAT_ORBS* and
*OCC_TOLERANCE*, or by prepending a valid method name with “fno” in
the energy call as

```
energy('fno-ccsd(t)')
```

The FNOCC module in PSI4 supports several related many-body quantum chemistry methods, including the CCSD(T) and QCISD(T) methods, several orders of many-body perturbation theory (MP2-MP4), and a family methods related to the coupled electron pair approximation (CEPA).

The quadratic configuration interaction singles doubles (QCISD) method of Pople, Head-Gordon, and Raghavachari [Pople:1987:5968] was originally presented as a size-consistent extension of configuration interaction singles doubles (CISD). The method can also be obtained as a simplified version of the coupled cluster singles doubles (CCSD) method [Purvis:1982]. Consider the set of equations defining CCSD:

(1)

where we have chosen the intermediate normalization, , and the symbols and represent single and double excitation operators. The QCISD equations can be obtained by omitting all but two terms that are nonlinear in and :

(2)

QCISD is slightly cheaper that CCSD computationally, but it retains the complexity of the original equations. Just as in the familiar CCSD(T) method, the effects of connected triple excitations may be included noniteratively to yield the QCISD(T) method. Both the QCISD(T) and CCSD(T) methods are implemented for closed-shell references in the FNOCC module.

QCI and CC methods are closely related to perturbation theory, and the
MP2, MP3, and MP4(SDQ) correlation energies can be obtained as a free
by-product of a CCSD or QCISD computation. The following is an
example of the results for a computation run with the call
`energy('fno-qcisd')` to `energy()`:

```
QCISD iterations converged!
OS MP2 FNO correction: -0.000819116338
SS MP2 FNO correction: -0.000092278158
MP2 FNO correction: -0.000911394496
OS MP2 correlation energy: -0.166478414245
SS MP2 correlation energy: -0.056669079827
MP2 correlation energy: -0.223147494072
* MP2 total energy: -76.258836941658
OS MP2.5 correlation energy: -0.171225850256
SS MP2.5 correlation energy: -0.054028401038
MP2.5 correlation energy: -0.225254251294
* MP2.5 total energy: -76.260943698880
OS MP3 correlation energy: -0.175973286267
SS MP3 correlation energy: -0.051387722248
MP3 correlation energy: -0.227361008515
* MP3 total energy: -76.263050456101
OS MP4(SDQ) correlation energy: -0.180324322304
SS MP4(SDQ) correlation energy: -0.048798468084
MP4(SDQ) correlation energy: -0.230995119324
* MP4(SDQ) total energy: -76.266684566910
OS QCISD correlation energy: -0.181578117924
SS QCISD correlation energy: -0.049853548145
QCISD correlation energy: -0.231431666069
* QCISD total energy: -76.267121113654
```

The first set of energies printed corresponds to the second-order FNO correction mentioned previously. Results for many-body perturbation theory through partial fourth order are then provided. The notation MP4(SDQ) indicates that we have included all contributions to the correlation energy through fourth order, with the exception of that from connected triple excitations.

One need not run a full QCISD or CCSD computation to obtain these
perturbation theory results. The keywords for invoking perturbation
theory computations are given below in
Table *FNOCC Methods*. Full MP4 correlation
energies are also available.

Coupled-pair methods can be viewed as approximations to CCSD or as size-extensive modifications of CISD. The methods have the same complexity as CISD, and solving the CISD or coupled-pair equations requires fewer floating point operations than solving the CCSD. CISD, CCSD, and the coupled-pair methods discussed below all scale formally with the sixth power of system size, and, as with the QCISD method, CEPA methods retain complexity of the CCSD equations. For a detailed discussion of the properties of various coupled-pair methods, see Ref. [Wennmohs:2008:217].

What follows is a very basic description of the practical differences in the equations that define each of the coupled-pair methods implemented in PSI4. We begin with the CISD wave function

(3)

where we have chosen the intermediate normalization, . The CISD correlation energy is given by

(4)

and the amplitudes can be determined by the solution to the coupled set of eqations:

(5)

The CISD method is not size-extensive, but this problem can be overcome by making very simple modifications to the amplitude equations. We replace the correlation energy, , with generalized shifts for the doubles and singles equations, and :

(6)

These shifts approximate the effects of triple and quadruple excitations.
The values for and used in several
coupled-pair methods are given in Table *CEPA Shifts*. Note that these shifts are defined in a spin-free
formalism for closed-shell references only.

method sdci dci NA cepa(0) 0 0 cepa(1) cepa(3) acpf aqcc

The pair correlation energy, , is simply a partial sum of the correlation energy. In a spin-free formalism, the pair energy is given by

(7)

Methods whose shifts ( and ) do not explicitly depend on orbitals or (CISD, CEPA(0), ACPF, and AQCC) have solutions that render the energy stationary with respect variations in the amplitudes. This convenient property allows density matrices and 1-electron properties to be evaluated without any additional effort. Note, however, that 1-electron properties are currently unavailable when coupling these stationary CEPA-like methods with frozen natural orbitals.

Density fitting (DF) or resolution of the identity (RI) techniques are popular in quantum chemistry to avoid the computation and storage of the 4-index electron repulsion integral (ERI) tensor and even to reduce the computational scaling of some terms. DF-CCSD(T) computations are available in PSI4, with or without the use of FNOs, through the FNOCC module. The implementation and accuracy of the DF-CCSD(T) method are described in Ref. [DePrince:2013:inprep].

The default auxiliary basis set for a DF-CCSD computation is chosen to be the RI set (optimized for DFMP2) most similar to the primary basis set. For example, if the primary basis set is aug-cc-pVDZ, the default auxiliary basis set will be the aug-cc-pVDZ-RI set. PSI4 of course allows the user to specify any supported predefined basis set as the auxiliary set. Alternatively, the user can request a set defined by the partial Cholesky decomposition of the 4-index ERI tensor.

The following is a minimal input file that describes a DF-CCSD(T) computation using 3-index integrals obtained by partial Cholesky decomposition of the 4-index ERI tensor.

```
molecule h2o {
0 1
O
H 1 1.0
H 1 1.0 2 104.5
}
set {
df_basis_cc cholesky
basis aug-cc-pvdz
freeze_core true
}
energy('df-ccsd(t)')
```

The accuracy of the Cholesky decomposition may be controlled through the
keyword *CHOLESKY_TOLERANCE*. Note that the keyword
*SCF_TYPE* has not been specified here. By default, a DF-CCSD(T)
computation exploits DF technology in the SCF procedure, but one can
override this behavior through this keyword.

The FNOCC module contains all the components that comprise the Gn family
of composite methods. Currently, only the G2 method is supported
[Curtiss:1991:7221]. The G2 procedure may be called through the
`energy()` wrapper:

```
energy('gaussian-2')
```

The various methods supported by the FNOCC module in PSI4 are detailed
in Table *FNOCC Methods*. Note that these methods
are implemented for closed-shell references only. For open-shell references,
the calls `energy('mp2.5')`, `energy('mp3')`, and `energy('mp4')` will
default to the *DETCI* implementations of these methods.

name calls method qcisd quadratic configuration interaction singles doubles qcisd(t) qcisd with perturbative triples mp2.5 average of second- and third-order perturbation theories mp3 third-order perturbation theory mp4(sdq) fourth-order perturbation theory, minus triples contribution mp4 full fourth-order perturbation theory cepa(0) coupled electron pair approximation, variant 0 cepa(1) coupled electron pair approximation, variant 1 cepa(3) coupled electron pair approximation, variant 3 acpf averaged coupled-pair functional aqcc averaged quadratic coupled-cluster sdci configuration interaction with single and double excitations dci configuration interaction with double excitations fno-qcisd qcisd with frozen natural orbitals fno-qcisd(t) qcisd(t) with frozen natural orbitals fno-ccsd coupled cluster singles doubles with frozen natural orbitals fno-ccsd(t) ccsd with perturbative triples and frozen natural orbitals fno-mp3 mp3 with frozen natural orbitals fno-mp4(sdq) mp4(sdq) with frozen natural orbitals fno-mp4 mp4 with frozen natural orbitals fno-cepa(0) cepa(0) with frozen natural orbitals fno-cepa(1) cepa(1) with frozen natural orbitals fno-cepa(3) cepa(3) with frozen natural orbitals fno-acpf acpf with frozen natural orbitals fno-aqcc aqcc with frozen natural orbitals fno-sdci sdci with frozen natural orbitals fno-dci dci with frozen natural orbitals df-ccsd ccsd with density fitting df-ccsd(t) ccsd(t) with density fitting fno-df-ccsd ccsd with density fitting and frozen natural orbitals fno-df-ccsd(t) ccsd(t) with density fitting and frozen natural orbitals

Primary basis set

Type: stringPossible Values:basis stringDefault: No Default

Specifies how many core orbitals to freeze in correlated computations.

TRUEwill default to freezing the standard default number of core orbitals. For PSI, the standard number of core orbitals is the number of orbitals in the nearest previous noble gas atom. More precise control over the number of frozen orbitals can be attained by using the keywordsNUM_FROZEN_DOCC(gives the total number of orbitals to freeze, program picks the lowest-energy orbitals) orFROZEN_DOCC(gives the number of orbitals to freeze per irreducible representation)

Type: stringPossible Values: FALSE, TRUEDefault: FALSE

Convergence for the CC amplitudes. Note that convergence is met only when

E_CONVERGENCEandR_CONVERGENCEare satisfied.

Type:conv doubleDefault: 1.0e-7

Convergence criterion for CC energy. See Table

Post-SCF Convergencefor default convergence criteria for different calculation types. Note that convergence is met only whenE_CONVERGENCEandR_CONVERGENCEare satisfied.

Type:conv doubleDefault: 1.0e-8

Desired number of DIIS vectors

Type: integerDefault: 8

Do use MP2 NOs to truncate virtual space for QCISD/CCSD and (T)?

Type:booleanDefault: false

Cutoff for occupation of MP2 NO orbitals in FNO-QCISD/CCSD(T) ( only valid if

NAT_ORBS= true )

Type:conv doubleDefault: 1.0e-6

Do use low memory option for triples contribution? Note that this option is enabled automatically if the memory requirements of the conventional algorithm would exceed the available resources

Type:booleanDefault: false

Do time each cc diagram?

Type:booleanDefault: false

Auxilliary basis for df-ccsd(t).

Type: stringPossible Values:basis stringDefault: No Default

tolerance for Cholesky decomposition of the ERI tensor

Type:conv doubleDefault: 1.0e-4

Flag to exclude singly excited configurations from a coupled-pair computation.

Type:booleanDefault: false

Opposite-spin scaling factor for SCS-MP2

Type: doubleDefault: 1.20

Same-spin scaling factor for SCS-MP2

Type: doubleDefault: 1.0/3.0

Oppposite-spin scaling factor for SCS-CCSD

Type: doubleDefault: 1.27

Same-spin scaling factor for SCS-CCSD

Type: doubleDefault: 1.13

Is this a CEPA job? This parameter is used internally by the pythond driver. Changing its value won’t have any effect on the procedure.

Type:booleanDefault: false

Do compute triples contribution?

Type:booleanDefault: true

Do compute MP4 triples contribution?

Type:booleanDefault: false

Do use density fitting in CC? This keyword is used internally by the driver. Changing its value will have no effect on the computation.

Type:booleanDefault: false

Which coupled-pair method is called? This parameter is used internally by the python driver. Changing its value won’t have any effect on the procedure.

Type: stringDefault: CEPA(0)