.. include:: autodoc_abbr_options_c.rst .. _`sec:cepa`: .. index:: Coupled-Pair Methods,CEPA .. index:: pair: Coupled-Pair Methods; theory pair: CEPA; theory CEPA: Coupled-Pair Methods ========================== .. codeauthor:: A. Eugene DePrince .. sectionauthor:: A. Eugene DePrince *Module:* :ref:`Keywords `, :ref:`PSI Variables `, :source:`CEPA ` Coupled-pair methods can be viewed as approximations to coupled-cluster (CC) theory or as size-extensive modifications of truncated configuration interaction (CI) theory. The methods have the same complexity as CI with single and double excitations (CISD), and solving the CISD or coupled-pair equations requires fewer floating point operations than solving the CC with singles and doubles (CCSD) equations. CISD, CCSD, and the coupled-pair methods discussed below all scale formally with the sixth power of system size. For a detailed discussion of the properties of various coupled-pair methods, see Ref. [Wennmohs:2008]_\ . What follows is a very basic description of the practical differences in the equations that define each of the coupled-pair methods implemented in |Psifour|. We begin with the CISD wave function .. math:: :label: CIwfn | \Psi \rangle = | \Psi_0 \rangle + \sum_i^{occ} \sum_a^{vir} t_i^a | \Psi_i^a\rangle + \frac{1}{4}\sum_{ij}^{occ} \sum_{ab}^{vir} t_{ij}^{ab} | \Psi_{ij}^{ab}\rangle, where we have chosen the intermediate normalization, :math:`\langle \Psi_0 | \Psi \rangle = 1`. The CISD correlation energy is given by .. math:: :label: CIenergy E_c = \langle \Psi_0 | \hat{H} - E_0 | \Psi \rangle, and the amplitudes can be determined by the solution to the coupled set of eqations: .. math:: :label: CIeqns 0 &= \langle \Psi_{ij}^{ab} | \hat{H} - E_0 - E_c | \Psi \rangle, \\ 0 &= \langle \Psi_{i}^{a} | \hat{H} - E_0 - E_c | \Psi \rangle. The CISD method is not size-extensive, but this problem can be overcome by making very simple modifications to the amplitude equations. With malice and forethought, we replace the correlation energy, :math:`E_c`, with generalized shifts for the doubles and singles equations, :math:`\Delta_{ij}` and :math:`\Delta_i`: .. math:: :label: CEPAeqns 0 &= \langle \Psi_{ij}^{ab} | \hat{H} - E_0 - \Delta_{ij} | \Psi \rangle, \\ 0 &= \langle \Psi_{i}^{a} | \hat{H} - E_0 - \Delta_i | \Psi \rangle. These shifts approximate the effects of triple and quadruple excitations. The values for :math:`\Delta_{ij}` and :math:`\Delta_i` used in several coupled-pair methods are given in Table :ref:`CEPA Shifts `. Note that these shifts are defined in a spin-free formalism for closed-shell references only. .. _`table:cepa_shifts`: +-------------------------+------------------------------------------------------------+----------------------------------------------+ | method | :math:`\Delta_{ij}` | :math:`\Delta_i` | +=========================+============================================================+==============================================+ | sdci | :math:`E_c` | :math:`E_c` | +-------------------------+------------------------------------------------------------+----------------------------------------------+ | dci | :math:`E_c` | NA | +-------------------------+------------------------------------------------------------+----------------------------------------------+ | cepa(0) | 0 | 0 | +-------------------------+------------------------------------------------------------+----------------------------------------------+ | cepa(1) | :math:`\frac{1}{2}\sum_k(\epsilon_{ik}+\epsilon_{jk})` | :math:`\sum_k \epsilon_{ik}` | +-------------------------+------------------------------------------------------------+----------------------------------------------+ | cepa(3) | :math:`-\epsilon_{ij}+\sum_k(\epsilon_{ik}+\epsilon_{jk})` | :math:`-\epsilon_{ii}+2\sum_k \epsilon_{ik}` | +-------------------------+------------------------------------------------------------+----------------------------------------------+ | acpf | :math:`\frac{2}{N} E_c` | :math:`\frac{2}{N} E_c` | +-------------------------+------------------------------------------------------------+----------------------------------------------+ | aqcc | :math:`[1-\frac{(N-3)(N-2)}{N(N-1)}]E_c` | :math:`[1-\frac{(N-3)(N-2)}{N(N-1)}]E_c` | +-------------------------+------------------------------------------------------------+----------------------------------------------+ The pair correlation energy, :math:`\epsilon_{ij}`, is simply a partial sum of the correlation energy. In a spin-free formalism, the pair energy is given by .. math:: :label: pair_energy \epsilon_{ij} = \sum_{ab} v_{ij}^{ab} (2 t_{ij}^{ab} - t_{ij}^{ba}) Methods whose shifts (:math:`\Delta_{ij}` and :math:`\Delta_i`) do not explicitly depend on orbitals :math:`i` or :math:`j` (CISD, CEPA(0), ACPF, and AQCC) have solutions that render the energy stationary with respect variations in the amplitudes. This convenient property allows density matrices and 1-electron properties to be evaluated without any additional effort. The coupled-pair methods currently supported in |Psifour| are outlined in Table :ref:`CEPA Methods `. .. _`table:cepa_calls`: +-------------------------+--------------------------------------------------------------+---------+-------------+------------------------+ | name | calls method | energy | derivatives | 1-electron properties | +=========================+==============================================================+=========+=============+========================+ | cepa(0) | coupled electron pair approximation, variant 0 | Y | N | Y | +-------------------------+--------------------------------------------------------------+---------+-------------+------------------------+ | cepa(1) | coupled electron pair approximation, variant 1 | Y | N | N | +-------------------------+--------------------------------------------------------------+---------+-------------+------------------------+ | cepa(3) | coupled electron pair approximation, variant 3 | Y | N | N | +-------------------------+--------------------------------------------------------------+---------+-------------+------------------------+ | acpf | averaged coupled-pair functional | Y | N | Y | +-------------------------+--------------------------------------------------------------+---------+-------------+------------------------+ | aqcc | averaged quadratic coupled-cluster | Y | N | Y | +-------------------------+--------------------------------------------------------------+---------+-------------+------------------------+ | sdci | configuration interaction with single and double excitations | Y | N | Y | +-------------------------+--------------------------------------------------------------+---------+-------------+------------------------+ | dci | configuration interaction with double excitations | Y | N | Y | +-------------------------+--------------------------------------------------------------+---------+-------------+------------------------+ .. index:: CEPA; basic-keywords Basic Coupled Pair Keywords ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. include:: /autodir_options_c/mints__basis.rst .. include:: /autodir_options_c/globals__freeze_core.rst .. include:: /autodir_options_c/cepa__r_convergence.rst .. include:: /autodir_options_c/cepa__maxiter.rst .. include:: /autodir_options_c/cepa__diis_max_vecs.rst .. include:: /autodir_options_c/cepa__mp2_scale_os.rst .. include:: /autodir_options_c/cepa__mp2_scale_ss.rst .. include:: /autodir_options_c/cepa__dipmom.rst .. include:: /autodir_options_c/cepa__cepa_vabcd_direct.rst .. index:: CEPA; advanced-keywords Advanced Coupled Pair Keywords ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. include:: /autodir_options_c/cepa__cepa_level.rst .. include:: /autodir_options_c/cepa__scs_cepa.rst .. include:: /autodir_options_c/cepa__cepa_scale_os.rst .. include:: /autodir_options_c/cepa__cepa_scale_ss.rst .. include:: /autodir_options_c/cepa__cepa_no_singles.rst