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

Code author: Ugur Bozkaya

Section author: Ugur Bozkaya

Module: Keywords, PSI Variables, DFOCC

Introduction

Orbital-optimized methods have several advantages over 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 invarianti [Pedersen:1999:od].

Another advantage is that the orbital-optimized methods does not suffer from artifactual symmetry-breaking instabilities [Crawford:1997:instability], [Sherrill:1998:od], [Bozkaya:2011:omp2], and [Bozkaya:2011:omp3]. Further, 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 (hence Orbital optimized MP3) will solve this N-representability problem by disregarding orbital response contribution of one-partical 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.

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

\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

where \hat{K} is the orbital rotation operator

\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})

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

\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

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

\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}}

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.

Publications resulting from the use of the OMP2 code should cite the following publications:

[Bozkaya:2011:omp2] and [Bozkaya:2013:omp2grad].

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 we can simply use “reference rohf” option. For DFT orbitals one should use “reference uks” and “dft_functional b3lyp” options. Of course users can use any DFT functional available in Psi4.

Methods

Density-fitted conventional and orbital-optimized MP2 methods currently supported in PSI4 are outlined in Table DF-OMP2 Methods.

Name Calls Method Energy Gradient Reference
ri-mp2 Density-Fitted MP2 Y Y RHF/ROHF/UHF
cd-mp2 Cholesky-Decomposed MP2 Y N RHF/ROHF/UHF
df-omp2 Density-Fitted Orbital-Optimized MP2 Y Y RHF/ROHF/UHF/RKS/UKS
cd-omp2 Cholesky-Decomposed Orbital-Optimized MP2 Y N RHF/ROHF/UHF/RKS/UKS

Basic 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. Default adjusts depending on E_CONVERGENCE

MAX_MOGRAD_CONVERGENCE

Convergence criterion for maximum orbital gradient

MO_MAXITER

Maximum number of iterations to determine the orbitals

  • Type: integer
  • Default: 50

ORB_OPT

Do optimize the orbitals?

Advanced Keywords

OPT_METHOD

The orbital optimization algorithm. Presently Quasy Newton-Raphson algorithm avaliable 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?

DF-OCC: Conventional Møller–Plesset Perturbation Theories

Module: Keywords, PSI Variables, DFOCC

PSI4 also has a density-fitted MP2 algorithm for RHF, UHF, and ROHF energies in the DFOCC module.

Name Calls Method Energy Gradient Reference
ri-mp2 DF-MP2 Y Y RHF/ROHF/UHF
cd-mp2 MP2 Y N RHF/ROHF/UHF