# SAPT: Symmetry-Adapted Perturbation Theory

Code author: Edward G. Hohenstein, Rob M. Parrish, Jérôme F. Gonthier, and Daniel. G. A. Smith

Section author: Edward G. Hohenstein and Jérôme F. Gonthier

Module: Keywords, PSI Variables, LIBSAPT_SOLVER

Warning

In rare cases with systems having a high degree of symmetry, PSI4 gives (very obviously) wrong answers for SAPT computations when the specification is in Z-matrix format. Use a Cartesian representation to avoid this problem.

Caution

In early versions (notably PSI4 alpha circa 2011 and before), frozen core was implemented incompletely and for only selected terms. Comparisons with papers published using early PSI4 SAPT code may show discrepancies of 0.01-0.10 kcal/mol in individual terms, particularly \(E_{exch}^{(11)}\) and \(E_{exch}^{(12)}\).

Caution

January 28th 2016, the default for all NAT_ORBS options was changed to true. Hence the code now by default uses natural orbital truncation to speed up the evaluation of energy terms wherever possible, according to literature recommendations. In early July 2016, some total SAPT energy psivars were renamed.

Symmetry-adapted perturbation theory (SAPT) provides a means of directly computing the noncovalent interaction between two molecules, that is, the interaction energy is determined without computing the total energy of the monomers or dimer. In addition, SAPT provides a decomposition of the interaction energy into physically meaningful components: i.e., electrostatic, exchange, induction, and dispersion terms. In SAPT, the Hamiltonian of the dimer is partitioned into contributions from each monomer and the interaction.

\[H=F_A+W_A+F_B+W_B+V\]

Here, the Hamiltonian is written as a sum of the usual monomer Fock operators, \(F\), the fluctuation potential of each monomer, \(W\), and the interaction potential, \(V\). The monomer Fock operators, \(F_A+F_B\), are treated as the zeroth-order Hamiltonian and the interaction energy is evaluated through a perturbative expansion of \(V\), \(W_A\), and \(W_B\). Through first-order in \(V\), electrostatic and exchange interactions are included; induction and dispersion first appear at second-order in \(V\). For a complete description of SAPT, the reader is referred to the excellent review by Jeziorski, Moszynski, and Szalewicz [Jeziorski:1994:1887].

Several truncations of the closed-shell SAPT expansion are available in the SAPT module of PSI4. The simplest truncation of SAPT is denoted SAPT0 and defined in Eq. (1).

(1)\[E_{SAPT0} = E_{elst}^{(10)} + E_{exch}^{(10)} + E_{ind,resp}^{(20)} + E_{exch-ind,resp}^{(20)} + E_{disp}^{(20)} + E_{exch-disp}^{(20)} + \delta_{HF}^{(2)}\]

In this notation, \(E^{(vw)}\) defines the order in \(V\) and in \(W_A+W_B\); the subscript, \(resp\), indicates that orbital relaxation effects are included.

(2)\[E_{SAPT2} = E_{SAPT0} + E_{elst,resp}^{(12)} + E_{exch}^{(11)} + E_{exch}^{(12)} + \; ^{t}\!E_{ind}^{(22)} + \; ^{t}\!E_{exch-ind}^{(22)}\]
(3)\[E_{SAPT2+} = E_{SAPT2} + E_{disp}^{(21)} + E_{disp}^{(22)}\]
(4)\[E_{SAPT2+(3)} = E_{SAPT2+} + E_{elst,resp}^{(13)} + E_{disp}^{(30)}\]
(5)\[E_{SAPT2+3} = E_{SAPT2+(3)} + E_{exch-ind}^{(30)} + E_{ind,resp}^{(30)} + E_{exch-disp}^{(30)} + E_{ind-disp}^{(30)} + E_{exch-ind-disp}^{(30)} - \delta_{HF}^{(2)} + \delta_{HF}^{(3)}\]

The \(\delta_{HF}^{(2)}\) and \(\delta_{HF}^{(3)}\) terms take into account higher-order induction effects and are included in the definition of SAPT terms. They are computed from the Hartree–Fock supermolecular interaction energy \(E_{int}^{HF}\) and are only available in dimer-centered basis SAPT computations, which is the default (see below for monomer-centered basis computations). They are defined by:

(6)\[\delta_{HF}^{(2)} = E_{int}^{HF} - (E_{elst}^{(10)} + E_{exch}^{(10)} + E_{ind,resp}^{(20)} + E_{exch-ind,resp}^{(20)})\]
(7)\[\delta_{HF}^{(3)} = \delta_{HF}^{(2)} - (E_{exch-ind}^{(30)} + E_{ind,resp}^{(30)})\]

Additionally, high-order coupling between induction and dispersion can be extracted from the supermolecular MP2 interaction energy:

\[\delta_{MP2}^{(2)} = E_{int}^{MP2, corr} - (E_{elst}^{(12)} + E_{exch}^{(11)} + E_{exch}^{(12)} + \; ^{t}\!E_{ind}^{(22)} + \; ^{t}\!E_{exch-ind}^{(22)} + E_{disp}^{(20)} + E_{exch-disp}^{(20)})\]
\[\delta_{MP2}^{(3)} = \delta_{MP2}^{(2)} - (E_{ind-disp}^{(30)} + E_{exch-ind-disp}^{(30)})\]

where \(E_{int}^{MP2, corr}\) is the correlation part of the supermolecular MP2 interaction energy. \(\delta_{MP2}^{(2)}\) and \(\delta_{MP2}^{(3)}\) also improve the description of electrostatically dominated complexes. \(\delta_{MP2}^{(2)}\) can be applied to SAPT2+ or SAPT2+(3) energies whereas \(\delta_{MP2}^{(3)}\) should be applied to SAPT2+3 energies.

A thorough analysis of the performance of these truncations of closed-shell SAPT can be found in a review by Hohenstein and Sherrill [Hohenstein:2012:WIREs], and a systematic study of the accuracy of these truncations (with and without an improved CCD treatment of dispersion) using different basis sets is reported in [Parker:2014:094106].

The closed-shell SAPT module relies entirely on the density-fitting approximation of the two-electron integrals. The factorization of the SAPT energy expressions, as implemented in PSI4, assumes the use of density-fitted two-electron integrals, therefore, the closed-shell SAPT module cannot be run with exact integrals. In practice, we have found that the density-fitting approximation introduces negligible errors into the SAPT energy (often less than 0.01 kcal/mol for small dimers) and greatly improves efficiency.

The latest addition to the SAPT code is the SAPT0 method for open-shell monomers [Gonthier:2016:134106]. This code is available for both exact and density fitted integrals, except for the dispersion terms which implementation relies on a density fitting factorization. Both `UHF` and `ROHF` REFERENCE can be used, but coupled induction computations are currently not supported with `ROHF`. This means that orbital relaxation is not included for `ROHF` and the uncoupled induction term is computed instead. If both monomers are open-shell, their coupling is assumed to be high spin, i.e. two doublets would interact to form a triplet.

## The S2 approximation and scaling

All exchange terms in SAPT arise from the antisymmetrization of the wavefunctions of monomers A and B. Taking into account exchange of all possible electron pairs between the two monomers yields to complicated formulae. For this reason, exchange terms are often evaluated in the \(S^{2}\) approximation, that can be interpreted as the exchange of a single electron pair between monomers.

The \(S^{2}\) approximation is usually pretty good, but may break down for short intermolecular distance, particularly in high-order terms. To compensate these deviations, Parker et al. [Parker:2014:094106] recommend to scale all \(S^{2}\) approximated exchange terms by the ratio:

\[p_{EX}(\alpha) = \left( \frac{E_{exch}^{(10)}}{E_{exch}^{(10)}(S^{2})} \right)^{\alpha}\]

where the recommended exponent is \(\alpha = 1\). To obtain SAPT energies with this scaling, simply set the keyword `exch_scale_alpha true`. Alternatively, another value for \(\alpha\) can be specified by setting EXCH_SCALE_ALPHA to a value. For example,

    1
    set exch_scale_alpha 1.0

will set \(\alpha = 1.0\) and scale exchange energies with \(p_{EX}(1.0)\).

Instead of this straightforward scaling, SAPT0 energies benefit from a slightly modified recipe that involves an empirically adjusted exponent \(\alpha = 3.0\). To distinguish it from its unscaled counterpart, this energy is denoted sSAPT0 (see [Parker:2014:094106]).

(8)\[E_{sSAPT0} = E_{elst}^{(10)} + E_{exch}^{(10)} + E_{ind,resp}^{(20)} + p_{EX}(3.0) E_{exch-ind,resp}^{(20)} + E_{disp}^{(20)} + p_{EX}(3.0) E_{exch-disp}^{(20)} + \delta_{HF}^{(2)}\]

where \(\delta_{HF}^{(2)}\) is computed without any scaling. Please note that sSAPT0 is thus not the same as requesting `exch_scale_alpha 3.0`, and that the scaling is automatically performed by requesting `energy('ssapt0')`.

## A First Example

The following is the simplest possible input that will perform all available SAPT computations (normally, you would pick one of these methods).

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    molecule water_dimer {
         0 1
         O  -1.551007  -0.114520   0.000000
         H  -1.934259   0.762503   0.000000
         H  -0.599677   0.040712   0.000000
         --
         0 1
         O   1.350625   0.111469   0.000000
         H   1.680398  -0.373741  -0.758561
         H   1.680398  -0.373741   0.758561

         units angstrom
         no_reorient
         symmetry c1
    }

    set basis aug-cc-pvdz

    energy('sapt0')
    energy('sapt2')
    energy('sapt2+')
    energy('sapt2+(3)')
    energy('sapt2+3')

The SAPT module uses the standard PSI4 partitioning of the dimer into monomers. SAPT does not use spatial symmetry and needs the geometry of the system to remain fixed throughout monomer and dimer calculations. These requirements are imposed whenever a SAPT calculation is requested but can also be set explicitly with the `no_reorient` and `symmetry c1` molecule keywords, as in the example above. As a reminder, only SAPT0 can handle the interaction of both closed- and open-shell monomers. Higher-order SAPT is only available for computation of interactions between closed-shell singlets.

The example input shown above would not be used in practice. To exploit the efficiency of the density-fitted SAPT implementation in PSI4, the SCF computations should also be performed with density-fitted (DF) integrals.

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    11
    set globals {
        basis         aug-cc-pvdz
        df_basis_scf  aug-cc-pvdz-jkfit
        df_basis_sapt aug-cc-pvdz-ri
        guess         sad
        scf_type      df
    }

    set sapt {
        print         1
    }

These options will perform the SAPT computation with DF-HF and a superposition-of-atomic-densities guess. This is the preferred method of running the SAPT module. As already mentioned above, the SAPT0 module for open-shell cases can also use exact integrals for all terms except for dispersion. In practice, density fitting is considerably faster and introduces negligible errors, thus it is the preferred method for open-shell cases as well.

Below, you can find a minimum example of open-shell SAPT0 computation.

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    molecule {
         0 1
         O 0.000000  0.000000  6.000000
         H 0.000000  1.431500  4.890600
         H 0.000000 -1.431500  4.890600
         --
         0 2
         O 0.000000  0.000000  0.000000
         O 0.000000  2.503900  0.000000
         H 0.000000 -0.424700 -1.839500
         units bohr
         symmetry c1
         no_reorient
         no_com
    }

    set {
    reference uhf
    scf_type     df
    basis         cc-pVDZ
    }

    energy('sapt0')

REFERENCE needs to be `UHF` or `ROHF` for the open-shell computation to proceed.

## Advanced example

Open-shell computations can be difficult to converge in certain cases, thus you may want to have more control over the SCF procedure. You have the option of doing the driver job in the input file, by performing the dimer and monomer computations yourself. In the example below, we do a stability analysis for the open-shell monomer only

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
    50
    51
    52
    53
    54
    55
    56
    57
    58
    59
    molecule {
         0 2
         O 0.000000  0.000000  0.000000
         O 0.000000  2.503900  0.000000
         H 0.000000 -0.424700 -1.839500
         --
         0 1
         O 0.000000  0.000000  6.000000
         H 0.000000  1.431500  4.890600
         H 0.000000 -1.431500  4.890600
         units bohr
         symmetry c1
         no_reorient
         no_com
    }

    dimer = psi4.get_active_molecule()

    set {
    reference uhf
    scf_type     df
    basis         cc-pVDZ
    df_basis_sapt cc-pVDZ-ri
    guess sad
    }

    dimer = psi4.get_active_molecule()

    set df_ints_io save
    psi4.IO.set_default_namespace('dimer')
    Edim, wfn_dimer = energy('scf',molecule=dimer,return_wfn=True)
    set df_ints_io load

    monomerA = dimer.extract_subsets(1,2)
    psi4.IO.change_file_namespace(97, 'dimer', 'monomerA')
    psi4.IO.set_default_namespace('monomerA')
    set {
    stability_analysis follow
    }
    EmonA, wfn_monA = energy('scf',molecule=monomerA,return_wfn=True)

    monomerB = dimer.extract_subsets(2,1)
    psi4.IO.change_file_namespace(97, 'monomerA', 'monomerB')
    psi4.IO.set_default_namespace('monomerB')
    set {
    stability_analysis none
    }
    EmonB, wfn_monB = energy('scf',molecule=monomerB,return_wfn=True)

    psi4.IO.change_file_namespace(97, 'monomerB', 'dimer')
    psi4.IO.set_default_namespace('dimer')

    aux_basis = psi4.core.BasisSet.build(wfn_dimer.molecule(), "DF_BASIS_SAPT",
                                  psi4.core.get_global_option("DF_BASIS_SAPT"),
                                  "RIFIT", psi4.core.get_global_option("BASIS"))
    wfn_dimer.set_basisset("DF_BASIS_SAPT", aux_basis)
    wfn_dimer.set_basisset("DF_BASIS_ELST", aux_basis)

    psi4.sapt(wfn_dimer,wfn_monA,wfn_monB)

In this way, any of the SCF options can be tweaked for individual fragments. For optimal efficiency, the example uses `set df_ints_io save` to keep file 97, which contains the three-index integrals for density fitting. `set df_ints_io load` then instructs the program to read these integrals from disk instead of recomputing them. For each SCF computation, we use `psi4.IO.set_default_namespace` to uniquely name scratch files. In the following SCF step, only file 97 is renamed using `psi4.IO.change_file_namespace` so that integrals can be read from it. For more information on stability analysis, see the stability documentation.

## SAPT0

Generally speaking, SAPT0 should be applied to large systems or large data sets. The performance of closed-shell SAPT0 relies entirely on error cancellation, which seems to be optimal with a truncated aug-cc-pVDZ basis, namely, jun-cc-pVDZ (which we have referred to in previous work as aug-cc-pVDZ’). We do not recommend using SAPT0 with large basis sets like aug-cc-pVTZ. A systematic study of the accuracy of closed-shell SAPT0 and other SAPT truncations, using different basis sets, is reported in [Parker:2014:094106]. In particular, an empirical recipe for scaled SAPT0 can yield improved performance and has been included in the output file as the sSAPT0 interaction energy. sSAPT0 is a free by-product and is automatically computed when SAPT0 is requested (see above for more details). The SAPT module has been used to perform SAPT0 computations with over 200 atoms and 2800 basis functions; this code should be scalable to 4000 basis functions. Publications resulting from the use of the SAPT0 code should cite the following publications: [Hohenstein:2010:184111] and [Hohenstein:2011:174107]. If the open-shell SAPT0 code is used, [Gonthier:2016:134106] should be additionally cited.

### Basic SAPT0 Keywords

#### SAPT_LEVEL

> The level of theory for SAPT

> - Type: string
> - Possible Values: SAPT0, SAPT2, SAPT2+, SAPT2+3
> - Default: SAPT0

#### BASIS

> Primary basis set, describes the monomer molecular orbitals

> - Type: string
> - Possible Values: basis string
> - Default: No Default

#### DF_BASIS_SAPT

> Auxiliary basis set for SAPT density fitting computations. Defaults to a RI basis.

> - Type: string
> - Possible Values: basis string
> - Default: No Default

#### DF_BASIS_ELST

> Auxiliary basis set for SAPT Elst10 and Exch10 density fitting computations, may be important if heavier elements are involved. Defaults to DF_BASIS_SAPT

> - Type: string
> - Possible Values: basis string
> - Default: No Default

#### FREEZE_CORE

> The scope of core orbitals to freeze in evaluation of SAPT \(E_{disp}^{(20)}\) and \(E_{exch-disp}^{(20)}\) terms. Recommended true for all SAPT computations

> - Type: string
> - Possible Values: FALSE, TRUE
> - Default: FALSE

#### D_CONVERGENCE

> Convergence criterion for residual of the CPHF coefficients in the SAPT \(E_{ind,resp}^{(20)}\) term.

> - Type: conv double
> - Default: 1e-8

#### E_CONVERGENCE

> Convergence criterion for energy (change) in the SAPT \(E_{ind,resp}^{(20)}\) term during solution of the CPHF equations.

> - Type: conv double
> - Default: 1e-10

#### MAXITER

> Maximum number of CPHF iterations

> - Type: integer
> - Default: 50

#### PRINT

> The amount of information to print to the output file for the sapt module. For 0, only the header and final results are printed. For 1, (recommended for large calculations) some intermediate quantities are also printed.

> - Type: integer
> - Default: 1

### Advanced SAPT0 Keywords

#### AIO_CPHF

> Do use asynchronous disk I/O in the solution of the CPHF equations? Use may speed up the computation slightly at the cost of spawning an additional thread.

> - Type: boolean
> - Default: false

#### AIO_DF_INTS

> Do use asynchronous disk I/O in the formation of the DF integrals? Use may speed up the computation slightly at the cost of spawning an additional thread.

> - Type: boolean
> - Default: false

#### COUPLED_INDUCTION

> Solve the CPHF equations to compute coupled induction and exchange-induction. These are not available for ROHF, and the option is automatically false in this case. In all other cases, coupled induction is strongly recommended. Only turn it off if the induction energy is not going to be used.

> - Type: boolean
> - Default: true

#### EXCH_SCALE_ALPHA

> Whether or not to perform exchange scaling for SAPT exchange components. Default is false, i.e. no scaling. If set to true, performs scaling with \(Exch10 / Exch10(S^2)\). If set to a value \(\alpha\), performs scaling with \((Exch10 / Exch10(S^2))^{\alpha}\).

> - Type: string
> - Default: FALSE

#### INTS_TOLERANCE

> Schwarz screening threshold. Minimum absolute value below which all three-index DF integrals and those contributing to four-index integrals are neglected. The default is conservative, but there isn’t much to be gained from loosening it, especially for higher-order SAPT.

> - Type: conv double
> - Default: 1.0e-12

#### DENOMINATOR_DELTA

> Maximum error allowed (Max error norm in Delta tensor) in the approximate energy denominators employed for most of the \(E_{disp}^{(20)}\) and \(E_{exch-disp}^{(20)}\) evaluation.

> - Type: double
> - Default: 1.0e-6

#### DENOMINATOR_ALGORITHM

> Denominator algorithm for PT methods. Laplace transformations are slightly more efficient.

> - Type: string
> - Possible Values: LAPLACE, CHOLESKY
> - Default: LAPLACE

#### DEBUG

> The amount of information to print to the output file

> - Type: integer
> - Default: 0

### Specific open-shell SAPT0 keywords

#### SAPT_MEM_SAFETY

> Memory safety

> - Type: double
> - Default: 0.9

#### COUPLED_INDUCTION

> Solve the CPHF equations to compute coupled induction and exchange-induction. These are not available for ROHF, and the option is automatically false in this case. In all other cases, coupled induction is strongly recommended. Only turn it off if the induction energy is not going to be used.

> - Type: boolean
> - Default: true

## Higher-Order SAPT

For smaller systems (up to the size of a nucleic acid base pair), more accurate interaction energies can be obtained through higher-order SAPT computations. The SAPT module can perform density-fitted evaluations of SAPT2, SAPT2+, SAPT2+(3), and SAPT2+3 energies for closed-shell systems only. Publications resulting from the use of the higher-order SAPT code should cite the following: [Hohenstein:2010:014101].

For methods SAPT2+ and above, one can replace the many-body treatment of dispersion by an improved method based on coupled-cluster doubles (CCD). This approach tends to give good improvements when dispersion effects are very large, as in the PCCP dimer (see [Hohenstein:2011:2842]). As shown in [Parker:2014:094106], whether or not CCD dispersion offers more accurate interaction energies tends to depend on the SAPT truncation and basis set employed, due to cancellations of errors. Thanks to natural orbital methods [Parrish:2013:174102], the SAPT code is able to include CCD dispersion with only a modest additional cost. Computations employing CCD dispersion should cite [Parrish:2013:174102]. To request CCD dispersion treatment in a SAPT computation, simply append `(ccd)` to the name of the method, as in the following examples

    1
    2
    3
    energy('sapt2+(ccd)')
    energy('sapt2+(3)(ccd)')
    energy('sapt2+3(ccd)')

The \(\delta_{MP2}\) corrections can also be computed automatically by appending `dmp2` to the name of the method, with or without CCD dispersion

    1
    2
    3
    4
    5
    6
    energy('sapt2+dmp2')
    energy('sapt2+(3)dmp2')
    energy('sapt2+3dmp2')
    energy('sapt2+(ccd)dmp2')
    energy('sapt2+(3)(ccd)dmp2')
    energy('sapt2+3(ccd)dmp2')

A brief note on memory usage: the higher-order SAPT code assumes that certain quantities can be held in core. This code requires sufficient memory to hold \(3o^2v^2+v^2N_{aux}\) arrays in core. With this requirement computations on the adenine-thymine complex can be performed with an aug-cc-pVTZ basis in less than 64GB of memory.

Higher-order SAPT is treated separately from the highly optimized SAPT0 code, therefore, higher-order SAPT uses a separate set of keywords. The following keywords are relevant for higher-order SAPT.

### Basic Keywords for Higher-order SAPT

#### BASIS

> Primary basis set, describes the monomer molecular orbitals

> - Type: string
> - Possible Values: basis string
> - Default: No Default

#### DF_BASIS_SAPT

> Auxiliary basis set for SAPT density fitting computations. Defaults to a RI basis.

> - Type: string
> - Possible Values: basis string
> - Default: No Default

#### FREEZE_CORE

> Specifies how many core orbitals to freeze in correlated computations. `TRUE` or `1` will default to freezing the previous noble gas shell on each atom. In case of positive charges on fragments, an additional shell may be unfrozen, to ensure there are valence electrons in each fragment. With `FALSE` or `0`, no electrons are frozen (with the exception of electrons treated by an ECP). With `-1`, `-2`, and `-3`, the user might request strict freezing of the previous first/second/third noble gas shell on every atom. In this case, when there are no valence electrons, the code raises an exception. More precise control over the number of frozen orbitals can be attained by using the keywords NUM_FROZEN_DOCC (gives the total number of orbitals to freeze, program picks the lowest-energy orbitals) or FROZEN_DOCC (gives the number of orbitals to freeze per irreducible representation)

> - Type: string
> - Possible Values: FALSE, TRUE, 1, 0, -1, -2, -3
> - Default: FALSE

#### PRINT

> The amount of information to print to the output file for the sapt module. For 0, only the header and final results are printed. For 1, (recommended for large calculations) some intermediate quantities are also printed.

> - Type: integer
> - Default: 1

### Advanced Keywords for Higher-order SAPT

#### DO_CCD_DISP

> Do CCD dispersion correction in SAPT2+, SAPT2+(3) or SAPT2+3?

> - Type: boolean
> - Default: false

#### DO_MBPT_DISP

> Do MBPT dispersion correction in SAPT2+, SAPT2+(3) or SAPT2+3, if also doing CCD?

> - Type: boolean
> - Default: true

#### DO_THIRD_ORDER

> Do compute third-order corrections?

> - Type: boolean
> - Default: false

#### INTS_TOLERANCE

> Schwarz screening threshold. Minimum absolute value below which all three-index DF integrals and those contributing to four-index integrals are neglected. The default is conservative, but there isn’t much to be gained from loosening it, especially for higher-order SAPT.

> - Type: conv double
> - Default: 1.0e-12

#### SAPT_MEM_CHECK

> Do force SAPT2 and higher to die if it thinks there isn’t enough memory? Turning this off is ill-advised.

> - Type: boolean
> - Default: true

#### DEBUG

> The amount of information to print to the output file

> - Type: integer
> - Default: 0

## MP2 Natural Orbitals

One of the unique features of the SAPT module is its ability to use MP2 natural orbitals (NOs) to speed up the evaluation of the triples contribution to dispersion. By transforming to the MP2 NO basis, we can throw away virtual orbitals that are expected to contribute little to the dispersion energy. Speedups in excess of \(50 \times\) are possible. In practice, this approximation is very good and should always be applied. Publications resulting from the use of MP2 NO-based approximations should cite the following: [Hohenstein:2010:104107].

### Basic Keywords Controlling MP2 NO Approximations

#### NAT_ORBS_T2

> Do use MP2 natural orbital approximations for the \(v^4\) block of two-electron integrals in the evaluation of second-order T2 amplitudes? Recommended true for all SAPT computations.

> - Type: boolean
> - Default: true

#### NAT_ORBS_T3

> Do natural orbitals to speed up evaluation of the triples contribution to dispersion by truncating the virtual orbital space? Recommended true for all SAPT computations.

> - Type: boolean
> - Default: true

#### NAT_ORBS_V4

> Do use MP2 natural orbital approximations for the \(v^4\) block of two-electron integrals in the evaluation of CCD T2 amplitudes? Recommended true for all SAPT computations.

> - Type: boolean
> - Default: true

#### OCC_TOLERANCE

> Minimum occupation (eigenvalues of the MP2 OPDM) below which virtual natural orbitals are discarded for in each of the above three truncations

> - Type: conv double
> - Default: 1.0e-6

## Charge-Transfer in SAPT

It is possible to obtain the stabilization energy of a complex due to charge-transfer effects from a SAPT computation. The charge-transfer energy can be computed with the SAPT module as described by Stone and Misquitta [Misquitta:2009:201].

Charge-transfer energies can be obtained from the following calls to the energy function.

    1
    2
    3
    4
    5
    6
    7
    8
    energy('sapt0-ct')
    energy('sapt2-ct')
    energy('sapt2+-ct')
    energy('sapt2+(3)-ct')
    energy('sapt2+3-ct')
    energy('sapt2+(ccd)-ct')
    energy('sapt2+(3)(ccd)-ct')
    energy('sapt2+3(ccd)-ct')

For now, charge transfer computations are not available with open-shell SAPT0.

A SAPT charge-transfer analysis will perform 5 HF computations: the dimer in the dimer basis, monomer A in the dimer basis, monomer B in the dimer basis, monomer A in the monomer A basis, and monomer B in the monomer B basis. Next, it performs two SAPT computations, one in the dimer basis and one in the monomer basis. Finally, it will print a summary of the charge-transfer results:

    1
    2
    3
    4
    5
      SAPT Charge Transfer Analysis
    ------------------------------------------------------------------------------------------------
      SAPT Induction (Dimer Basis)       -2.0970 [mEh]      -1.3159 [kcal/mol]      -5.5057 [kJ/mol]
      SAPT Induction (Monomer Basis)     -1.1396 [mEh]      -0.7151 [kcal/mol]      -2.9920 [kJ/mol]
      SAPT Charge Transfer               -0.9574 [mEh]      -0.6008 [kcal/mol]      -2.5137 [kJ/mol]

These results are for the water dimer geometry shown above computed with SAPT0/aug-cc-pVDZ.

## Monomer-Centered Basis Computations

The charge-transfer analysis above is carried out by taking the difference between SAPT induction as calculated in the dimer-centered basis (i.e., each monomer sees the basis functions on both monomers) vs. the monomer-centered basis (i.e., each monomer utilizes only its own basis set). It is also possible to run a closed-shell SAPT computation at any level using only the monomer-centered basis. To do this, simply add `sapt_basis='monomer'` to the energy function, such as

    1
    energy('sapt2',sapt_basis='monomer')

This procedure leads to faster compuations, but it converges more slowly towards the complete basis set limit than the default procedure, which uses the dimer-centered basis set. Hence, monomer-centered basis SAPT computations are not recommended. The open-shell SAPT0 code is not compatible yet with monomer-centered computations.

## Computations with Mid-bonds

SAPT computations with midbonds can be accomplished by adding a third ghost monomer to the computation. For example

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    molecule dimer {
        0 1
        He 0  0  5
        --
        0 1
        He 0  0 -5
        --
        0 1
        @He 0  0  0
    }

Here the functions of the third monomer will be added to the virtual space of the entire computation. Note that an error will be thrown if each atom in the third monomer is not a ghost to prevent confusion with three-body SAPT which is not currently supported by Psi4.

## Interpreting SAPT Results

We will examine the results of a SAPT2+3/aug-cc-pVDZ computation on the water dimer. This computation can be performed with the following input:

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    molecule water_dimer {
         0 1
         O  -1.551007  -0.114520   0.000000
         H  -1.934259   0.762503   0.000000
         H  -0.599677   0.040712   0.000000
         --
         0 1
         O   1.350625   0.111469   0.000000
         H   1.680398  -0.373741  -0.758561
         H   1.680398  -0.373741   0.758561
         units angstrom
    }

    set globals {
        basis          aug-cc-pvdz
        guess          sad
        scf_type       df
    }

    set sapt {
        print          1
        nat_orbs_t2    true
        freeze_core    true
    }

    energy('sapt2+3')

To reiterate some of the options mentioned above: the NAT_ORBS_T2 option will compute MP2 natural orbitals and use them in the evaluation of the triples correction to dispersion, and the FREEZE_CORE option will freeze the core throughout the SAPT computation. This SAPT2+3/aug-cc-pVDZ computation produces the following results:

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
      SAPT Results
    --------------------------------------------------------------------------------------------------------
      Electrostatics                -13.06509118 [mEh]      -8.19846883 [kcal/mol]     -34.30239689 [kJ/mol]
        Elst10,r                    -13.37542977 [mEh]      -8.39320925 [kcal/mol]     -35.11719087 [kJ/mol]
        Elst12,r                      0.04490350 [mEh]       0.02817737 [kcal/mol]       0.11789413 [kJ/mol]
        Elst13,r                      0.26543510 [mEh]       0.16656305 [kcal/mol]       0.69689985 [kJ/mol]

      Exchange                       13.41768202 [mEh]       8.41972294 [kcal/mol]      35.22812415 [kJ/mol]
        Exch10                       11.21822294 [mEh]       7.03954147 [kcal/mol]      29.45344432 [kJ/mol]
        Exch10(S^2)                  11.13802706 [mEh]       6.98921779 [kcal/mol]      29.24289005 [kJ/mol]
        Exch11(S^2)                   0.04558907 [mEh]       0.02860757 [kcal/mol]       0.11969410 [kJ/mol]
        Exch12(S^2)                   2.15387002 [mEh]       1.35157390 [kcal/mol]       5.65498573 [kJ/mol]

      Induction                      -3.91313050 [mEh]      -2.45552656 [kcal/mol]     -10.27392413 [kJ/mol]
        Ind20,r                      -4.57530818 [mEh]      -2.87104935 [kcal/mol]     -12.01247162 [kJ/mol]
        Ind30,r                      -4.91714746 [mEh]      -3.08555675 [kcal/mol]     -12.90997067 [kJ/mol]
        Ind22                        -0.83718642 [mEh]      -0.52534243 [kcal/mol]      -2.19803293 [kJ/mol]
        Exch-Ind20,r                  2.47828501 [mEh]       1.55514739 [kcal/mol]       6.50673730 [kJ/mol]
        Exch-Ind30,r                  4.33916119 [mEh]       2.72286487 [kcal/mol]      11.39246770 [kJ/mol]
        Exch-Ind22                    0.45347471 [mEh]       0.28455969 [kcal/mol]       1.19059785 [kJ/mol]
        delta HF,r (2)               -1.43239563 [mEh]      -0.89884187 [kcal/mol]      -3.76075473 [kJ/mol]
        delta HF,r (3)               -0.85440936 [mEh]      -0.53614999 [kcal/mol]      -2.24325177 [kJ/mol]

      Dispersion                     -3.62000698 [mEh]      -2.27158877 [kcal/mol]      -9.50432831 [kJ/mol]
        Disp20                       -3.54291925 [mEh]      -2.22321549 [kcal/mol]      -9.30193450 [kJ/mol]
        Disp30                        0.05959979 [mEh]       0.03739944 [kcal/mol]       0.15647926 [kJ/mol]
        Disp21                        0.11216169 [mEh]       0.07038252 [kcal/mol]       0.29448051 [kJ/mol]
        Disp22 (SDQ)                 -0.17892163 [mEh]      -0.11227502 [kcal/mol]      -0.46975875 [kJ/mol]
        Disp22 (T)                   -0.47692534 [mEh]      -0.29927518 [kcal/mol]      -1.25216749 [kJ/mol]
        Est. Disp22 (T)              -0.54385233 [mEh]      -0.34127251 [kcal/mol]      -1.42788430 [kJ/mol]
        Exch-Disp20                   0.64545587 [mEh]       0.40502969 [kcal/mol]       1.69464439 [kJ/mol]
        Exch-Disp30                  -0.01823410 [mEh]      -0.01144207 [kcal/mol]      -0.04787362 [kJ/mol]
        Ind-Disp30                   -0.91816882 [mEh]      -0.57615966 [kcal/mol]      -2.41065224 [kJ/mol]
        Exch-Ind-Disp30               0.76487181 [mEh]       0.47996433 [kcal/mol]       2.00817094 [kJ/mol]

    Total HF                         -5.68662563 [mEh]      -3.56841161 [kcal/mol]     -14.93023559 [kJ/mol]
    Total SAPT0                      -8.58408901 [mEh]      -5.38659740 [kcal/mol]     -22.53752571 [kJ/mol]
    Total SAPT2                      -6.72343814 [mEh]      -4.21902130 [kcal/mol]     -17.65238683 [kJ/mol]
    Total SAPT2+                     -7.33405042 [mEh]      -4.60218631 [kcal/mol]     -19.25554938 [kJ/mol]
    Total SAPT2+(3)                  -7.00901553 [mEh]      -4.39822383 [kcal/mol]     -18.40217026 [kJ/mol]
    Total SAPT2+3                    -7.18054663 [mEh]      -4.50586123 [kcal/mol]     -18.85252518 [kJ/mol]

    Special recipe for scaled SAPT0 (see Manual):
      Electrostatics sSAPT0         -13.37542977 [mEh]      -8.39320925 [kcal/mol]     -35.11719087 [kJ/mol]
      Exchange sSAPT0                11.21822294 [mEh]       7.03954147 [kcal/mol]      29.45344432 [kJ/mol]
      Induction sSAPT0               -3.47550008 [mEh]      -2.18090932 [kcal/mol]      -9.12492546 [kJ/mol]
      Dispersion sSAPT0              -2.88342055 [mEh]      -1.80937379 [kcal/mol]      -7.57042064 [kJ/mol]
    Total sSAPT0                     -8.51612746 [mEh]      -5.34395089 [kcal/mol]     -22.35909265 [kJ/mol]
    --------------------------------------------------------------------------------------------------------

At the bottom of this output are the total SAPT energies (defined above), they are composed of subsets of the individual terms printed above. The individual terms are grouped according to the component of the interaction to which they contribute. The total component energies (i.e., electrostatics, exchange, induction, and dispersion) represent what we regard as the best estimate available at a given level of SAPT computed from a subset of the terms of that grouping. The groupings shown above are not unique and are certainly not rigorously defined. We regard the groupings used in PSI4 as a “chemist’s grouping” as opposed to a more mathematically based grouping, which would group all exchange terms (i.e. \(E_{exch-ind,resp}^{(20)}\), \(E_{exch-disp}^{(20)}\), etc.) in the exchange component. A final note is that both `Disp22(T)` and `Est.Disp22(T)` results appear if MP2 natural orbitals are used to evaluate the triples correction to dispersion. The `Disp22(T)` result is the triples correction as computed in the truncated NO basis; `Est.Disp22(T)` is a scaled result that attempts to recover the effect of the truncated virtual space and is our best estimate. The `Est.Disp22(T)` value is used in the SAPT energy and dispersion component (see [Hohenstein:2010:104107] for details). Finally, this part of the output file contains sSAPT0, a special scaling scheme of the SAPT0 energy that can yield improved results and was described in more details above. The corresponding scaled total component energies are printed as well.

As mentioned above, SAPT results with scaled exchange are also optionally available by setting the EXCH_SCALE_ALPHA keyword. When activated, the unscaled results are printed first as reported above, and then repeated with exchange scaling for all relevant terms:

     1
     2
     3
     4
     5
     6
     7
     8
     9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
      SAPT Results ==> ALL S2 TERMS SCALED (see Manual) <==

      Scaling factor (Exch10/Exch10(S^2))^{Alpha} =     1.007200
      with Alpha =     1.000000
    --------------------------------------------------------------------------------------------------------
      Electrostatics                -13.06509118 [mEh]      -8.19846883 [kcal/mol]     -34.30239689 [kJ/mol]
        Elst10,r                    -13.37542977 [mEh]      -8.39320925 [kcal/mol]     -35.11719087 [kJ/mol]
        Elst12,r                      0.04490350 [mEh]       0.02817737 [kcal/mol]       0.11789413 [kJ/mol]
        Elst13,r                      0.26543510 [mEh]       0.16656305 [kcal/mol]       0.69689985 [kJ/mol]

      Exchange sc.                   13.43351854 [mEh]       8.42966050 [kcal/mol]      35.26970292 [kJ/mol]
        Exch10                       11.21822294 [mEh]       7.03954147 [kcal/mol]      29.45344432 [kJ/mol]
        Exch10(S^2)                  11.13802706 [mEh]       6.98921779 [kcal/mol]      29.24289005 [kJ/mol]
        Exch11(S^2) sc.               0.04591732 [mEh]       0.02881355 [kcal/mol]       0.12055592 [kJ/mol]
        Exch12(S^2) sc.               2.16937828 [mEh]       1.36130548 [kcal/mol]       5.69570268 [kJ/mol]

      Induction sc.                  -3.90986540 [mEh]      -2.45347768 [kcal/mol]     -10.26535160 [kJ/mol]
        Ind20,r                      -4.57530818 [mEh]      -2.87104935 [kcal/mol]     -12.01247162 [kJ/mol]
        Ind30,r                      -4.91714746 [mEh]      -3.08555675 [kcal/mol]     -12.90997067 [kJ/mol]
        Ind22                        -0.83718642 [mEh]      -0.52534243 [kcal/mol]      -2.19803293 [kJ/mol]
        Exch-Ind20,r sc.              2.49612913 [mEh]       1.56634474 [kcal/mol]       6.55358703 [kJ/mol]
        Exch-Ind30,r sc.              4.37040396 [mEh]       2.74247000 [kcal/mol]      11.47449560 [kJ/mol]
        Exch-Ind22 sc.                0.45673981 [mEh]       0.28660857 [kcal/mol]       1.19917038 [kJ/mol]
        delta HF,r (2) sc.           -1.45023975 [mEh]      -0.91003922 [kcal/mol]      -3.80760445 [kJ/mol]
        delta HF,r (3) sc.           -0.90349624 [mEh]      -0.56695248 [kcal/mol]      -2.37212939 [kJ/mol]

      Dispersion sc.                 -3.60998364 [mEh]      -2.26529903 [kcal/mol]      -9.47801205 [kJ/mol]
        Disp20                       -3.54291925 [mEh]      -2.22321549 [kcal/mol]      -9.30193450 [kJ/mol]
        Disp30                        0.05959979 [mEh]       0.03739944 [kcal/mol]       0.15647926 [kJ/mol]
        Disp21                        0.11216169 [mEh]       0.07038252 [kcal/mol]       0.29448051 [kJ/mol]
        Disp22 (SDQ)                 -0.17892163 [mEh]      -0.11227502 [kcal/mol]      -0.46975875 [kJ/mol]
        Disp22 (T)                   -0.47692534 [mEh]      -0.29927518 [kcal/mol]      -1.25216749 [kJ/mol]
        Est. Disp22 (T)              -0.54385233 [mEh]      -0.34127251 [kcal/mol]      -1.42788430 [kJ/mol]
        Exch-Disp20 sc.               0.65010327 [mEh]       0.40794598 [kcal/mol]       1.70684615 [kJ/mol]
        Exch-Disp30 sc.              -0.01836538 [mEh]      -0.01152445 [kcal/mol]      -0.04821832 [kJ/mol]
        Ind-Disp30                   -0.91816882 [mEh]      -0.57615966 [kcal/mol]      -2.41065224 [kJ/mol]
        Exch-Ind-Disp30 sc.           0.77037903 [mEh]       0.48342016 [kcal/mol]       2.02263015 [kJ/mol]

    Total HF                         -5.68662563 [mEh]      -3.56841161 [kcal/mol]     -14.93023559 [kJ/mol]
    Total SAPT0 sc.                  -8.57944161 [mEh]      -5.38368112 [kcal/mol]     -22.52532395 [kJ/mol]
    Total SAPT2 sc.                  -6.69968912 [mEh]      -4.20411857 [kcal/mol]     -17.59003378 [kJ/mol]
    Total SAPT2+ sc.                 -7.31030140 [mEh]      -4.58728357 [kcal/mol]     -19.19319632 [kJ/mol]
    Total SAPT2+(3) sc.              -6.98526650 [mEh]      -4.38332109 [kcal/mol]     -18.33981720 [kJ/mol]
    Total SAPT2+3 sc.                -7.15142168 [mEh]      -4.48758504 [kcal/mol]     -18.77605762 [kJ/mol]
    --------------------------------------------------------------------------------------------------------

The scaling factor is reported at the top (here `1.0072`) together with the \(\alpha\) parameter. All terms that are scaled are indicated by the `sc.` label. Note that if Exch10 is less than \(10^{-5}\), the scaling factor is set to \(1.0\).

Caution

To density fit the dispersion terms in SAPT, the RI auxiliary basis set (e.g., aug-cc-pVDZ-RI) controlled through DF_BASIS_SAPT performs well. For Fock-type terms (i.e., electrostatics, exchange, induction, and core Fock matrix elements in exchange-dispersion), the density-fitting auxiliary basis in the SAPT module (both SAPT0 and higher-order) is RI (more efficient for the small basis sets at which SAPT0 performs best) while the FISAPT module uses the more appropriate JKFIT (e.g., aug-cc-pVDZ-JKFIT). For heavier elements (i.e., second-row and beyond), the RI auxiliary basis is unsound for this role (insufficiently flexible). For SAPT0 in the SAPT module, a workaround is to set DF_BASIS_ELST (which controls Elst10 and Exch10 terms) to a JKFIT basis. For higher-order methods in SAPT module, there is no workaround; on-the-fly construction of an auxiliary basis through Cholesky decomposition (not implemented) is the long-term solution.

## Spin-Flip SAPT

SAPT0 with two open-shell references will always yield a high-spin complex. In order to obtain a SAPT-based estimate of the splittings between different spin states of a complex the first-order exchange energies for all multiplets can be shown to be a linear combination of two matrix elements: a diagonal exchange term that determines the spin-averaged effect and a spin-flip term responsible for the splittings between the states. The numerical factors in this linear combination are determined solely by the Clebsch-Gordan coefficients: accordingly, the \(S^{2}\) approximation implies a Heisenberg Hamiltonian picture with a single coupling strength parameter determining all the splittings. This method can be invoked with energy(“SF-SAPT”) and publications resulting from the use of the SF-SAPT code should cite the following publications: [Patkowski:2018:164110]