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# SAPT: Symmetry-Adapted Perturbation Theory

Code author: Edward G. Hohenstein

Section author: Edward G. Hohenstein

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.

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.

Here, the Hamiltonian is written as a sum of the usual monomer Fock operators, , the fluctuation potential of each monomer, , and the interaction potential, . The monomer Fock operators, , are treated as the zeroth-order Hamiltonian and the interaction energy is evaluated through a perturbative expansion of , , and . Through first-order in , electrostatic and exchange interactions are included; induction and dispersion first appear at second-order in . 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 SAPT expansion are available in the SAPT module of PSI4. The simplest truncation of SAPT is denoted SAPT0 and defined in Eq. (1).

(1)

In this notation, defines the order in and in ; the subscript, , indicates that orbital relaxation effects are included.

(2)

(3)

(4)

(5)

A thorough analysis of the performance of these truncations of SAPT can be found in a review by Hohenstein and Sherrill [Hohenstein:2012:WIREs].

The 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 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.

## 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).

    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 globals {
        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. A final note is that the SAPT module is only capable of performing SAPT computations for 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.

    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.

## SAPT0

Generally speaking, SAPT0 should be applied to large systems or large data sets. The performance of 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’). 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].

### 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 and 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 term.

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

#### E_CONVERGENCE

> Convergence criterion for energy (change) in the SAPT term during solution of the CPHF equations.

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

#### MAXITER

> Maxmum 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

#### NO_RESPONSE

> Don’t solve the CPHF equations? Evaluate and instead of their response-including coupterparts. Only turn on this option if the induction energy is not going to be used.

> - Type: boolean
> - Default: false

#### INTS_TOLERANCE

> 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 and 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

#### SAPT_OS_SCALE

> The scale factor used for opposite-spin pairs in SCS computations. SS/OS decomposition performed for and terms.

> - Type: double
> - Default: 6.0/5.0

#### SAPT_SS_SCALE

> The scale factor used for same-spin pairs in SCS computations. SS/OS decomposition performed for and terms.

> - Type: double
> - Default: 1.0/3.0

#### DEBUG

> The amount of information to print to the output file

> - Type: integer
> - Default: 0

## 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. Publications resulting from the use of the higher-order SAPT code should cite the following: [Hohenstein:2010:014101].

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 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 higly 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` will 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 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
> - 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

#### INTS_TOLERANCE

> 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 disperison. 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 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 block of two-electron integrals in the evaluation of second-order T2 amplitudes? Recommended true for all SAPT computations.

> - Type: boolean
> - Default: false

#### 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

### Advanced Keywords Controlling MP2 NO Approximations

## 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.

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

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:

      SAPT Charge Transfer Analysis
    -----------------------------------------------------------------------------
      SAPT Induction (Dimer Basis)         -2.0970 mH       -1.3159 kcal mol^-1
      SAPT Induction (Monomer Basis)       -1.1396 mH       -0.7151 kcal mol^-1
      SAPT Charge Transfer                 -0.9574 mH       -0.6008 kcal mol^-1

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 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

    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.

## 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:

    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:

      SAPT Results
    --------------------------------------------------------------------------
      Electrostatics            -13.06429805 mH      -8.19797114 kcal mol^-1
        Elst10,r                -13.37543274 mH      -8.39321111 kcal mol^-1
        Elst12,r                  0.04490253 mH       0.02817676 kcal mol^-1
        Elst13,r                  0.26623216 mH       0.16706321 kcal mol^-1

      Exchange                   13.41793548 mH       8.41988199 kcal mol^-1
        Exch10                   11.21823471 mH       7.03954885 kcal mol^-1
        Exch10(S^2)              11.13803867 mH       6.98922508 kcal mol^-1
        Exch11(S^2)               0.04558910 mH       0.02860760 kcal mol^-1
        Exch12(S^2)               2.15411167 mH       1.35172554 kcal mol^-1

      Induction                  -3.91333155 mH      -2.45565272 kcal mol^-1
        Ind20,r                  -4.57531220 mH      -2.87105187 kcal mol^-1
        Ind30,r                  -4.91715479 mH      -3.08556135 kcal mol^-1
        Ind22                    -0.83761074 mH      -0.52560870 kcal mol^-1
        Exch-Ind20,r              2.47828867 mH       1.55514969 kcal mol^-1
        Exch-Ind30,r              4.33916816 mH       2.72286924 kcal mol^-1
        Exch-Ind22                0.45370482 mH       0.28470409 kcal mol^-1
        delta HF,r (2)           -1.43240211 mH      -0.89884593 kcal mol^-1
        delta HF,r (3)           -0.85441547 mH      -0.53615383 kcal mol^-1

      Dispersion                 -3.62061213 mH      -2.27196851 kcal mol^-1
        Disp20                   -3.54292109 mH      -2.22321664 kcal mol^-1
        Disp30                    0.05959981 mH       0.03739945 kcal mol^-1
        Disp21                    0.11216179 mH       0.07038259 kcal mol^-1
        Disp22 (SDQ)             -0.17924270 mH      -0.11247650 kcal mol^-1
        Disp22 (T)               -0.47692549 mH      -0.29927528 kcal mol^-1
        Est. Disp22 (T)          -0.54385253 mH      -0.34127263 kcal mol^-1
        Exch-Disp20               0.64545652 mH       0.40503010 kcal mol^-1
        Exch-Disp30              -0.01823411 mH      -0.01144207 kcal mol^-1
        Ind-Disp30               -0.91816995 mH      -0.57616037 kcal mol^-1
        Exch-Ind-Disp30           0.76459013 mH       0.47978757 kcal mol^-1

      Total HF                   -5.68662366 mH      -3.56841037 kcal mol^-1
      Total SAPT0                -8.58408823 mH      -5.38659691 kcal mol^-1
      Total SAPT2                -6.72339084 mH      -4.21899163 kcal mol^-1
      Total SAPT2+               -7.26739725 mH      -4.56036082 kcal mol^-1
      Total SAPT2+(3)            -6.94156528 mH      -4.35589816 kcal mol^-1
      Total SAPT2+3              -7.11337921 mH      -4.46371303 kcal mol^-1

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. , , 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. The `Est.Disp22(T)` value used in the SAPT energy and dispersion component (see [Hohenstein:2010:104107] for details).

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