.. include:: autodoc_abbr_options_c.rst .. index:: single: Orbital-Optimized Methods, OMP2 single: Orbital-Optimized Methods, OMP3 .. index:: pair: Orbital-Optimized Methods; theory pair: OMP2; theory pair: OMP3; theory .. _`sec:ompn`: OMP\ *n*\ : Orbital-Optimized M\ |o_slash|\ ller--Plesset Perturbation Theory ============================================================================= .. codeauthor:: Ugur Bozkaya .. sectionauthor:: Ugur Bozkaya *Module:* :ref:`Keywords `, :ref:`PSI Variables `, :source:`OMP2 ` *Module:* :ref:`Keywords `, :ref:`PSI Variables `, :source:`OMP3 ` 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 the artifactual symmetry-breaking instabilities [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 M\ |o_slash|\ ller--Plesset perturbation theory as implemented in |Psifour|. 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 .. math:: \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 :math:`\hat{K}` is the orbital rotation operator .. math:: \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 .. math:: {\bf C({\bf \kappa})} = {\bf C^{(0)}} \ e^{{\bf K}} where :math:`{\bf C^{(0)}}` is the initial MO coefficient matrix and :math:`{\bf C({\bf \kappa})}` is the new MO coefficient matrix as a function of :math:`{\bf \kappa}`. Now, let us define a variational energy functional (Lagrangian) as a function of :math:`{\bf \kappa}` * OMP2 .. math:: \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 * OMP3 .. math:: \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 where :math:`\hat{H}^{\kappa}`, :math:`\hat{f}_{N}^{\kappa}`, and :math:`\hat{W}_{N}^{\kappa}` defined as .. math:: \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}} and first and second derivatives of the energy with respect to the :math:`{\bf \kappa}` parameter at :math:`{\bf \kappa} = 0` .. math:: w_{pq} = \frac{\partial \widetilde{E}}{\partial \kappa_{pq}} .. math:: 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 .. math:: \widetilde{E}^{(2)}({\bf \kappa}) = \widetilde{E}^{(0)} + {\bf \kappa^{\dagger} w} + \frac{1}{2}~{\bf \kappa^{\dagger} A \kappa} where :math:`{\bf w}` is the MO gradient vector, :math:`{\bf \kappa}` is the MO rotation vector, and :math:`{\bf A}` is the MO Hessian matrix. Therefore, minimizing the energy with respect to :math:`{\bf \kappa}` yields .. math:: {\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 publication(s): [Bozkaya:2011:omp2]_. Publications resulting from the use of the OMP3 code should cite the following publications: [Bozkaya:2011:omp2]_ and [Bozkaya:2011:omp3]_. Methods ~~~~~~~ The orbital-optimized MP2 methods currently supported in |Psifour| are outlined in Table :ref:`OMP2 Methods `. .. _`table:omp2_calls`: +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | Name | Calls Method | Energy | Gradient | Reference | +=========================+==============================================================+=========+==========+===================+ | omp2 | Orbital-Optimized MP2 | Y | Y | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | scs-omp2 | Spin-Component Scaled Orbital-Optimized MP2 | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | sos-omp2 | Spin-Opposite Scaled Orbital-Optimized MP2 | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | scsn-omp2 | A special version of SCS-OMP2 for nucleobase interactions | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | scs-mi-omp2 | A special version of SCS-OMP2 (from S22 database) | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | scs-omp2-vdw | A special version of SCS-OMP2 (from ethene dimers) | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | sos-pi-omp2 | A special version of SOS-OMP2 for :math:`\pi`-systems | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ The orbital-optimized MP3 methods currently supported in |Psifour| are outlined in Table :ref:`OMP3 Methods `. .. _`table:omp3_calls`: +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | Name | Calls Method | Energy | Gradient | Reference | +=========================+==============================================================+=========+==========+===================+ | omp3 | Orbital-Optimized MP3 | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | scs-omp3 | Spin-Component Scaled Orbital-Optimized MP3 | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | sos-omp3 | Spin-Opposite Scaled Orbital-Optimized MP3 | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | scsn-omp3 | A special version of SCS-OMP3 for nucleobase interactions | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | scs-mi-omp3 | A special version of SCS-OMP3 (from S22 database) | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | scs-omp3-vdw | A special version of SCS-OMP3 (from ethene dimers) | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ | sos-pi-omp3 | A special version of SOS-OMP3 for :math:`\pi`-systems | Y | N | RHF/UHF/RKS/UKS | +-------------------------+--------------------------------------------------------------+---------+----------+-------------------+ .. index:: OMP2; setting keywords .. index:: OMP3; setting keywords Basic Keywords ~~~~~~~~~~~~~~ .. include:: /autodir_options_c/omp2__e_convergence.rst .. include:: /autodir_options_c/omp2__r_convergence.rst .. include:: /autodir_options_c/omp2__rms_mograd_convergence.rst .. include:: /autodir_options_c/omp2__max_mograd_convergence.rst .. include:: /autodir_options_c/omp2__mo_maxiter.rst Advanced Keywords ~~~~~~~~~~~~~~~~~ .. include:: /autodir_options_c/omp2__opt_method.rst .. include:: /autodir_options_c/omp2__diis_max_vecs.rst .. include:: /autodir_options_c/omp2__lineq_solver.rst .. include:: /autodir_options_c/omp2__orth_type.rst .. include:: /autodir_options_c/omp2__mp2_os_scale.rst .. include:: /autodir_options_c/omp2__mp2_ss_scale.rst .. include:: /autodir_options_c/omp2__sos_scale.rst .. include:: /autodir_options_c/omp2__sos_scale2.rst .. include:: /autodir_options_c/omp2__nat_orbs.rst .. include:: /autodir_options_c/omp2__omp2_orbs_print.rst .. include:: /autodir_options_c/omp3__omp3_orbs_print.rst .. include:: /autodir_options_c/omp3__tpdm_abcd_type.rst