OCC: Orbital-Optimized Coupled-Cluster and Møller–Plesset Perturbation Theories

Code author: Ugur Bozkaya

Section author: Ugur Bozkaya

Module: Keywords, PSI Variables, OCC

Module: Keywords, PSI Variables, DFOCC

Introduction

Orbital-optimized methods have several advantages over their non-optimized counterparts. Once the orbitals are optimized, the wave function will obey the Hellmann–Feynman theorem for orbital rotation parameters. Therefore, there is no need for orbital response terms in the evaluation of analytic gradients. In other words, it is unnecessary to solve the first order coupled-perturbed CC and many-body perturbation theory (MBPT) equations. Further, computation of one-electron properties is easier because there are no response contributions to the particle density matrices (PDMs). Moreover, active space approximations can be readily incorporated into the CC methods [Krylov:2000:vod]. Additionally, orbital-optimized coupled-cluster avoids spurious second-order poles in its response function, and its transition dipole moments are gauge invariant [Pedersen:1999:od].

Another advantage is that the orbital-optimized methods do not suffer from artifactual symmetry-breaking instabilities [Crawford:1997:instability], [Sherrill:1998:od], [Bozkaya:2011:omp2], and [Bozkaya:2011:omp3]. Furthermore, Kurlancheek and Head-Gordon [Kurlancek:2009] demonstrated that first order properties such as forces or dipole moments are discontinuous along nuclear coordinates when such a symmetry breaking occurs. They also observed that although the energy appears well behaved, the MP2 method can have natural occupation numbers greater than 2 or less than 0, hence may violate the N-representability condition. They further discussed that the orbital response equations generally have a singularity problem at the unrestriction point where spin-restricted orbitals become unstable to unrestriction. This singularity yields to extremely large or small eigenvalues of the one-particle density matrix (OPDM). These abnormal eigenvalues may lead to unphysical molecular properties such as vibrational frequencies. However, orbital-optimized MP2 (also MP3) will solve this N-representability problem by disregarding orbital response contribution of one-particle density matrix.

Although the performance of coupled-cluster singles and doubles (CCSD) and orbital-optimized CCD (OD) is similar, the situation is different in the case of triples corrections, especially at stretched geometries [Bozkaya:2012:odtl]. Bozkaya and Schaefer demonstrated that orbital-optimized coupled cluster based triple corrections, especially those of asymmetrics, provide significantly better potential energy curves than CCSD based triples corrections.

A lot of the functionality in OCC has been enabled with Density Fitting (DF) and Cholesky Decomposition (CD) techniques, which can greatly speed up calculations and reduce memory requirements for typically negligible losses in accuracy.

NOTE: As will be discussed later, all methods with orbital-optimization functionality have non-orbital optimized counterparts. Consequently, there arise two possible ways to call density-fitted MP2. In most cases, users should prefer the DF-MP2 code described in the DF-MP2 section because it is faster. If gradients are needed (like in a geometry optimization), then the procedures outlined hereafter should be followed. In general, choose the desired method, reference, and ERI type (e.g., set reference uhf, set mp2_type df, opt('mp2')) and the most efficient module will be selected automatically, according to Cross-module Redundancies.

Thus, there arise a few categories of method, each with corresponding input keywords:

  • Orbital-optimized MP and CC methods with conventional integrals (OMP Methods OCC keywords)

  • Orbital-optimized MP and CC methods with DF and CD integrals (OMP Methods DFOCC keywords)

  • Non-orbital-optimized MP and CC methods with conventional integrals (MP/CC Methods OCC keywords)

  • Non-orbital-optimized MP and CC methods with DF and CD integrals (MP/CC Methods DFOCC keywords)

Theory

What follows is a very basic description of orbital-optimized Møller–Plesset perturbation theory as implemented in PSI4. We will follow our previous presentations ([Bozkaya:2011:omp2], [Bozkaya:2011:omp3], and [Bozkaya:2012:odtl])

The orbital variations may be expressed by means of an exponential unitary operator

\[\begin{split}\widetilde{\hat{p}}^{\dagger} &= e^{\hat{K}} \hat{p}^{\dagger} e^{-\hat{K}}\\ \widetilde{\hat{p}} &= e^{\hat{K}} \ \hat{p} \ e^{-\hat{K}} \\ | \widetilde{p} \rangle &= e^{\hat{K}} \ | p \rangle\end{split}\]

where \(\hat{K}\) is the orbital rotation operator

\[\begin{split}\hat{K} &= \sum_{p,q}^{} K_{pq} \ \hat{E}_{pq} = \sum_{p>q}^{} \kappa_{pq} \ \hat{E}_{pq}^{-} \\ \hat{E}_{pq} &= \hat{p}^{\dagger} \hat{q} \\ \hat{E}_{pq}^{-} &= \hat{E}_{pq} \ - \ \hat{E}_{qp} \\ {\bf K} &= Skew({\bf \kappa})\end{split}\]

The effect of the orbital rotations on the MO coefficients can be written as

\[{\bf C({\bf \kappa})} = {\bf C^{(0)}} \ e^{{\bf K}}\]

where \({\bf C^{(0)}}\) is the initial MO coefficient matrix and \({\bf C({\bf \kappa})}\) is the new MO coefficient matrix as a function of \({\bf \kappa}\). Now, let us define a variational energy functional (Lagrangian) as a function of \({\bf \kappa}\)

  • OMP2

\[\begin{split}\widetilde{E}({\bf \kappa}) &= \langle 0| \hat{H}^{\kappa} | 0 \rangle \\ &+ \langle 0| \big(\hat{W}_{N}^{\kappa}\hat{T}_{2}^{(1)}\big)_{c} | 0 \rangle \\ &+ \langle 0| \{\hat{\Lambda}_{2}^{(1)} \ \big(\hat{f}_{N}^{\kappa} \hat{T}_{2}^{(1)} \ + \ \hat{W}_{N}^{\kappa} \big)_{c}\}_{c} | 0 \rangle\end{split}\]
  • OMP3

\[\begin{split}\widetilde{E}({\bf \kappa}) &= \langle 0| \hat{H}^{\kappa} | 0 \rangle \\ &+ \langle 0| \big(\hat{W}_{N}^{\kappa}\hat{T}_{2}^{(1)}\big)_{c} | 0 \rangle \ + \ \langle 0| \big(\hat{W}_{N}^{\kappa}\hat{T}_{2}^{(2)}\big)_{c} | 0 \rangle \\ &+ \langle 0| \{\hat{\Lambda}_{2}^{(1)} \ \big(\hat{f}_{N}^{\kappa} \hat{T}_{2}^{(1)} \ + \ \hat{W}_{N}^{\kappa} \big)_{c}\}_{c} | 0 \rangle \\ &+ \langle 0| \{\hat{\Lambda}_{2}^{(1)} \ \big(\hat{f}_{N}^{\kappa} \hat{T}_{2}^{(2)} \ + \ \hat{W}_{N}^{\kappa}\hat{T}_{2}^{(1)} \big)_{c}\}_{c} | 0 \rangle \\ &+ \langle 0| \{\hat{\Lambda}_{2}^{(2)} \ \big(\hat{f}_{N}^{\kappa} \hat{T}_{2}^{(1)} \ + \ \hat{W}_{N}^{\kappa} \big)_{c}\}_{c} | 0 \rangle\end{split}\]
  • OLCCD

\[\begin{split}\widetilde{E}({\bf \kappa}) &= \langle 0| \hat{H}^{\kappa} | 0 \rangle \ + \ \langle 0| \big(\hat{W}_{N}^{\kappa}\hat{T}_{2}\big)_{c} | 0 \rangle \\ &+ \langle 0| \{\hat{\Lambda}_{2} \ \big(\hat{W}_{N}^{\kappa} \ + \ \hat{H}_{N}^{\kappa}\hat{T}_{2} \big)_{c}\}_{c} | 0 \rangle\end{split}\]

where subscript c means only connected diagrams are allowed, and \(\hat{H}^{\kappa}\), \(\hat{f}_{N}^{\kappa}\), and \(\hat{W}_{N}^{\kappa}\) defined as

\[\begin{split}\hat{H}^{\kappa} &= e^{-\hat{K}} \hat{H} e^{\hat{K}} \\ \hat{f}_{N}^{\kappa} &= e^{-\hat{K}} \hat{f}_{N}^{d} e^{\hat{K}} \\ \hat{W}_{N}^{\kappa} &= e^{-\hat{K}} \hat{W}_{N} e^{\hat{K}}\end{split}\]

where \(\hat{f}_{N}\), and \(\hat{W}_{N}\) are the one- and two-electron components of normal-ordered Hamiltonian. Then, first and second derivatives of the energy with respect to the \({\bf \kappa}\) parameter at \({\bf \kappa} = 0\)

\[w_{pq} = \frac{\partial \widetilde{E}}{\partial \kappa_{pq}}\]
\[A_{pq,rs} = \frac{\partial^2 \widetilde{E}}{\partial \kappa_{pq} \partial \kappa_{rs}}\]

Then the energy can be expanded up to second-order as follows

\[\widetilde{E}^{(2)}({\bf \kappa}) = \widetilde{E}^{(0)} + {\bf \kappa^{\dagger} w} + \frac{1}{2}~{\bf \kappa^{\dagger} A \kappa}\]

where \({\bf w}\) is the MO gradient vector, \({\bf \kappa}\) is the MO rotation vector, and \({\bf A}\) is the MO Hessian matrix. Therefore, minimizing the energy with respect to \({\bf \kappa}\) yields

\[{\bf \kappa} = -{\bf A^{-1}w}\]

This final equation corresponds to the usual Newton-Raphson step.

  • OREMP

The REMP hybrid perturbation theory is a constrained mixture of the Møller–Plesset perturbation theory and the Retaining the Excitation degree perturbation theory([Fink:2006:RE], [Behnle:2019:REMP]). The mixing ratio is determined by the parameter :math’:A:

\[\widehat{H}^{(0)}_\text{REMP} = (1-A)\widehat{H}^{(0)}_\text{RE} + A\widehat{H}^{(0)}_\text{MP}\]

Technically, the second order of RE corresponds to LCCD for RHF and UHF references. REMP2 and its orbital-optimized variant OREMP2 are thus straightforward to implement in a (O)LCCD program by appropriate scaling of residual vector contributions and density matrices.

Convergence Problems

For problematic open-shell systems, we recommend to use the ROHF or DFT orbitals as an initial guess for orbital-optimized methods. Both ROHF and DFT orbitals may provide better initial guesses than UHF orbitals, hence convergence may be significantly speeded up with ROHF or DFT orbitals. In order to use ROHF orbitals, simply set reference rohf. For DFT orbitals, set reference uks and set dft_functional b3lyp. Of course users can use any DFT functional available in PSI4.

Methods

The orbital-optimized MPn and OLCCD methods currently supported in PSI4 are outlined in Table Orbital-Optimzed OCC/DFOCC Methods. The following methods are available and can be controlled through OCC (conventional integrals CONV) and DFOCC (density-fitted DF and Cholesky-decomposed CD) keywords. Switching between the integrals treatments is controlled through “type select” values in the rightmost Table column.

Orbital-Optimized MP and LCCD capabilities of OCC/DFOCC modules

name

calls method

Energy

Gradient

type select

omp2

Orbital-Optimized MP2

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

MP2_TYPE CONV

Density-Fitted Orbital-Optimized MP2

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

MP2_TYPE DF

Cholesky-Decomposed Orbital-Optimized MP2

RHF/UHF/ROHF/RKS/UKS

MP2_TYPE CD

omp3

Orbital-Optimized MP3

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

MP_TYPE CONV

Density-Fitted Orbital-Optimized MP3

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

MP_TYPE DF

Cholesky-Decomposed Orbital-Optimized MP3

RHF/UHF/ROHF/RKS/UKS

MP_TYPE CD

omp2.5

Orbital-Optimized MP2.5

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

MP_TYPE CONV

Density-Fitted Orbital-Optimized MP2.5

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

MP_TYPE DF

Cholesky-Decomposed Orbital-Optimized MP2.5

RHF/UHF/ROHF/RKS/UKS

MP_TYPE CD

olccd

Orbital-Optimized Linear CCD

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

CC_TYPE CONV

Density-Fitted Orbital-Optimized LCCD

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

CC_TYPE DF

Cholesky-Decomposed Orbital-Optimized LCCD

RHF/UHF/ROHF/RKS/UKS

CC_TYPE CD

oremp2

Orbital-Optimized 2nd order REMP hybrid perturbation theory

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

CC_TYPE CONV

Density-Fitted Orbital-Optimized REMP

RHF/UHF/ROHF/RKS/UKS

RHF/UHF/ROHF/RKS/UKS

CC_TYPE DF

Cholesky-Decomposed Orbital-Optimized REMP

RHF/UHF/ROHF/RKS/UKS

CC_TYPE CD

Spin-Component-Scaled Orbital-Optimized MP capabilities of OCC/DFOCC modules

name

calls method

Energy

Gradient

scs-omp3

Spin-Component Scaled Orbital-Optimized MP3

RHF/UHF/ROHF/RKS/UKS

sos-omp3

Spin-Opposite Scaled Orbital-Optimized MP3

RHF/UHF/ROHF/RKS/UKS

scs(n)-omp3

A special version of SCS-OMP3 for nucleobase interactions

RHF/UHF/ROHF/RKS/UKS

scs-omp3-vdw

A special version of SCS-OMP3 (from ethene dimers)

RHF/UHF/ROHF/RKS/UKS

sos-pi-omp3

A special version of SOS-OMP3 for \(\pi\)-systems

RHF/UHF/ROHF/RKS/UKS

scs-omp2

Spin-Component Scaled Orbital-Optimized MP2

RHF/UHF/ROHF/RKS/UKS

sos-omp2

Spin-Opposite Scaled Orbital-Optimized MP2

RHF/UHF/ROHF/RKS/UKS

scs(n)-omp2

A special version of SCS-OMP2 for nucleobase interactions

RHF/UHF/ROHF/RKS/UKS

scs-omp2-vdw

A special version of SCS-OMP2 (from ethene dimers)

RHF/UHF/ROHF/RKS/UKS

sos-pi-omp2

A special version of SOS-OMP2 for \(\pi\)-systems

RHF/UHF/ROHF/RKS/UKS

Basic OCC Keywords

E_CONVERGENCE

Convergence criterion for energy. See Table Post-SCF Convergence for default convergence criteria for different calculation types.

R_CONVERGENCE

Convergence criterion for amplitudes (residuals).

RMS_MOGRAD_CONVERGENCE

Convergence criterion for RMS orbital gradient. If this keyword is not set by the user, OCC will estimate and use a value required to achieve the desired E_CONVERGENCE The listed default will be used for the default value of E_CONVERGENCE

MAX_MOGRAD_CONVERGENCE

Convergence criterion for maximum orbital gradient. If this keyword is not set by the user, OCC will estimate and use a value required to achieve the desired E_CONVERGENCE The listed default will be used for the default value of E_CONVERGENCE

MO_MAXITER

Maximum number of iterations to determine the orbitals

  • Type: integer

  • Default: 50

WFN_TYPE

Type of the wavefunction.

  • Type: string

  • Possible Values: OMP2, OMP3, OCEPA, OMP2.5, REMP, OREMP

  • Default: OMP2

ORB_OPT

Do optimize the orbitals?

Advanced OCC Keywords

OPT_METHOD

The optimization algorithm. Modified Steepest-Descent (MSD) takes a Newton-Raphson (NR) step with a crude approximation to diagonal elements of the MO Hessian. The ORB_RESP option obtains the orbital rotation parameters with a crude approximation to all elements of the MO Hessian. Additionally, for both methods a DIIS extrapolation will be performed with the DO_DIIS = TRUE option.

  • Type: string

  • Possible Values: MSD, ORB_RESP

  • Default: MSD

LINEQ_SOLVER

The solver will be used for simultaneous linear equations.

  • Type: string

  • Possible Values: CDGESV, FLIN, POPLE

  • Default: CDGESV

ORTH_TYPE

The algorithm for orthogonalization of MOs

  • Type: string

  • Possible Values: GS, MGS

  • Default: MGS

MP2_OS_SCALE

Removed in 1.4. Will raise an error in 1.5.

  • Type: double

  • Default: 6.0

MP2_SS_SCALE

Removed in 1.4. Will raise an error in 1.5.

  • Type: double

  • Default: 1.0

MP2_SOS_SCALE

Removed in 1.4. Will raise an error in 1.5.

  • Type: double

  • Default: 1.3

MP2_SOS_SCALE2

Removed in 1.4. Will raise an error in 1.5.

  • Type: double

  • Default: 1.2

NAT_ORBS

Do compute natural orbitals?

OCC_ORBS_PRINT

Do print OCC orbital energies?

TPDM_ABCD_TYPE

How to take care of the TPDM VVVV-block. The COMPUTE option means it will be computed via an IC/OOC algorithm. The DIRECT option (default) means it will not be computed and stored, instead its contribution will be directly added to Generalized-Fock Matrix.

  • Type: string

  • Possible Values: DIRECT, COMPUTE

  • Default: DIRECT

DO_DIIS

Do apply DIIS extrapolation?

DO_LEVEL_SHIFT

Removed in 1.4. Will raise an error in 1.5.

Basic DFOCC Keywords

E_CONVERGENCE

Convergence criterion for energy. See Table Post-SCF Convergence for default convergence criteria for different calculation types.

R_CONVERGENCE

Convergence criterion for amplitudes (residuals).

RMS_MOGRAD_CONVERGENCE

Convergence criterion for RMS orbital gradient. If this keyword is not set by the user, DFOCC will estimate and use a value required to achieve the desired E_CONVERGENCE The listed default will be used for the default value of E_CONVERGENCE

MAX_MOGRAD_CONVERGENCE

Convergence criterion for maximum orbital gradient. If this keyword is not set by the user, DFOCC will estimate and use a value required to achieve the desired E_CONVERGENCE The listed default will be used for the default value of E_CONVERGENCE

MO_MAXITER

Maximum number of iterations to determine the orbitals

  • Type: integer

  • Default: 100

ORB_OPT

Do optimize the orbitals?

Advanced DFOCC Keywords

OPT_METHOD

The orbital optimization algorithm. Presently quasi-Newton-Raphson algorithm available with several Hessian * options.

  • Type: string

  • Possible Values: QNR

  • Default: QNR

HESS_TYPE

Type of the MO Hessian matrix

  • Type: string

  • Possible Values: APPROX_DIAG, APPROX_DIAG_EKT, APPROX_DIAG_HF, HF

  • Default: HF

MO_DIIS_NUM_VECS

Number of vectors used in orbital DIIS

  • Type: integer

  • Default: 6

ORTH_TYPE

The algorithm for orthogonalization of MOs

  • Type: string

  • Possible Values: GS, MGS

  • Default: MGS

DO_DIIS

Do apply DIIS extrapolation?

DO_LEVEL_SHIFT

Do apply level shifting?

Conventional (Non-OO) Coupled-Cluster and Møller–Plesset Perturbation Theories

Non-orbital-optimized counterparts to higher order MPn methods are also available. The following methods are available and can be controlled through OCC (conventional integrals CONV) and DFOCC (density-fitted DF and Cholesky-decomposed CD) keywords. Switching between the integrals treatments is controlled through ‘type select’ values; see rightmost column in Table Conventional OCC/DFOCC Methods.

Depending on efficiency considerations, the OCC & DFOCC modules may or may not be the default in PSI4 for available methods. (See Cross-module Redundancies for gory details.) To call the OCC/DFOCC implementation of any method below in preference to the default module, issue set qc_module occ.

Starting in v1.4, MP2.5 and MP3 default to the density-fit algorithm. Set MP_TYPE to CONV to get previous behavior.

Publications resulting from the use of the non-OO CC codes should cite the following publications:

Conventional (non-OO) CC and MP capabilities of OCC/DFOCC modules

name

calls method

Energy

Gradient

type select

mp2

MP2

RHF/UHF/ROHF

RHF/UHF

MP2_TYPE CONV

Density-Fitted MP2

RHF/UHF/ROHF

RHF/UHF

MP2_TYPE DF

Cholesky-Decomposed MP2

RHF/UHF/ROHF

MP2_TYPE CD

mp3

MP3

RHF/UHF

RHF/UHF

MP_TYPE CONV

Density-Fitted MP3

RHF/UHF

RHF/UHF

MP_TYPE DF

Cholesky-Decomposed MP3

RHF/UHF

MP_TYPE CD

mp2.5

MP2.5

RHF/UHF

RHF/UHF

MP_TYPE CONV

Density-Fitted MP2.5

RHF/UHF

RHF/UHF

MP_TYPE DF

Cholesky-Decomposed MP2.5

RHF/UHF

MP_TYPE CD

lccd

Linearized CCD

RHF/UHF

RHF/UHF

CC_TYPE CONV

Density-Fitted LCCD

RHF/UHF

RHF/UHF

CC_TYPE DF

Cholesky-Decomposed LCCD

RHF/UHF

CC_TYPE CD

ccd

CCD

CC_TYPE CONV

Density-Fitted CCD

RHF

RHF

CC_TYPE DF

Cholesky-Decomposed CCD

RHF

CC_TYPE CD

ccsd

CCSD

CC_TYPE CONV

Density-Fitted CCSD

RHF

RHF

CC_TYPE DF

Cholesky-Decomposed CCSD

RHF

CC_TYPE CD

ccsd(t)

CCSD(T)

CC_TYPE CONV

Density-Fitted CCSD(T)

RHF

RHF

CC_TYPE DF

Cholesky-Decomposed CCSD(T)

RHF

CC_TYPE CD

a-ccsd(t)

Lambda-CCSD(T)

CC_TYPE CONV

Density-Fitted Lambda-CCSD(T)

RHF

CC_TYPE DF

Cholesky-Decomposed Lambda-CCSD(T)

RHF

CC_TYPE CD

remp2

2nd order REMP hybrid perturbation theory

RHF/UHF

CC_TYPE CONV

Density-Fitted REMP

RHF/UHF

CC_TYPE DF

Cholesky-Decomposed REMP

RHF/UHF

CC_TYPE CD