# 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.

Publications resulting from the use of the orbital-optimized code should cite the following publications:

- OMP2 [Bozkaya:2011:omp2] and [Bozkaya:2013:omp2grad]
- OMP3 [Bozkaya:2011:omp3] , [Bozkaya:2013:omp3], and [Bozkaya:2013:omp3grad]
- OMP2.5 [Bozkaya:2011:omp3]
- OLCCD [Bozkaya:2013:ocepa]
- LCCD [Bozkaya:2013:ocepa]

## 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

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.

> - Type: conv double
> - Default: 1e-6

### R_CONVERGENCE

> Convergence criterion for amplitudes (residuals).

> - Type: conv double
> - Default: 1e-5

### RMS_MOGRAD_CONVERGENCE

> Convergence criterion for RMS orbital gradient. Default adjusts depending on E_CONVERGENCE

> - Type: conv double
> - Default: 1e-6

### MAX_MOGRAD_CONVERGENCE

> Convergence criterion for maximum orbital gradient

> - Type: conv double
> - Default: 1e-3

### 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
> - Default: OMP2

### ORB_OPT

> Do optimize the orbitals?

> - Type: boolean
> - Default: true

## 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 by solving the orbital-reponse (coupled-perturbed CC) equations. Additionally, for both methods a DIIS extrapolation will be performed with the DO_DIIS = TRUE option.

> - Type: string
> - Possible Values: MSD, ORB_RESP
> - Default: ORB_RESP

### MO_DIIS_NUM_VECS

> Number of vectors used in orbital DIIS

> - Type: integer
> - Default: 6

### 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

> MP2 opposite-spin scaling value

> - Type: double
> - Default: 6.0

### MP2_SS_SCALE

> MP2 same-spin scaling value

> - Type: double
> - Default: 1.0

### MP2_SOS_SCALE

> MP2 Spin-opposite scaling (SOS) value

> - Type: double
> - Default: 1.3

### MP2_SOS_SCALE2

> Spin-opposite scaling (SOS) value for optimized-MP2 orbitals

> - Type: double
> - Default: 1.2

### NAT_ORBS

> Do compute natural orbitals?

> - Type: boolean
> - Default: false

### OCC_ORBS_PRINT

> Do print OCC orbital energies?

> - Type: boolean
> - Default: false

### 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?

> - Type: boolean
> - Default: true

### DO_LEVEL_SHIFT

> Do apply level shifting?

> - Type: boolean
> - Default: true

## Basic DFOCC Keywords

### E_CONVERGENCE

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

> - Type: conv double
> - Default: 1e-6

### R_CONVERGENCE

> Convergence criterion for amplitudes (residuals).

> - Type: conv double
> - Default: 1e-5

### RMS_MOGRAD_CONVERGENCE

> Convergence criterion for RMS orbital gradient. Default adjusts depending on E_CONVERGENCE

> - Type: conv double
> - Default: 1e-6

### MAX_MOGRAD_CONVERGENCE

> Convergence criterion for maximum orbital gradient

> - Type: conv double
> - Default: 1e-3

### MO_MAXITER

> Maximum number of iterations to determine the orbitals

> - Type: integer
> - Default: 50

### ORB_OPT

> Do optimize the orbitals?

> - Type: boolean
> - Default: true

## 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?

> - Type: boolean
> - Default: true

### DO_LEVEL_SHIFT

> Do apply level shifting?

> - Type: boolean
> - Default: true

## 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`.

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
ccsd(at) | Lambda-CCSD(T) | — | — | CC_TYPE CONV
Density-Fitted Lambda-CCSD(T) | RHF | — | CC_TYPE DF
Cholesky-Decomposed Lambda-CCSD(T) | RHF | — | CC_TYPE CD