.. include:: autodoc_abbr_options_c.rst .. index:: Frozen natural orbital coupled cluster, FNO-CC .. index:: single: QCISD(T) single: MP4 single: CEPA single: LCCSD single: Frozen Natural Orbitals single: FNO-QCISD(T) single: FNO-MP4 single: FNO-CCSD(T) single: DF-CCSD(T) pair: CCSD(T); density-fitting .. _`sec:fnocc`: FNOCC: Frozen natural orbitals for CCSD(T), QCISD(T), CEPA, and MP4 =================================================================== .. codeauthor:: A. Eugene DePrince .. sectionauthor:: A. Eugene DePrince *Module:* :ref:`Keywords `, :ref:`PSI Variables `, :source:`FNOCC ` .. warning:: There is a known bug concerning the i7-5930 series combined with the Intel 15 compilers and MKL 11.2.3. When |PsiFour| is compiled under these conditions, parallel runs of the FNOCC code have experienced non-sensical CCSD correlation energies (often several hartrees lower than the starting guess). At the moment, the only confirmed solutions are running serially, using a different BLAS implementation, or upgrading to Intel 16.0.2 and MKL 11.3.2. Frozen natural orbitals (FNO) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The computational cost of the CCSD [Purvis:1982]_, CCSD(T) [Raghavachari:1989]_, and related methods be reduced by constructing a compact representation of the virtual space based on the natural orbitals of second-order perturbation theory [Sosa:1989:148]_. The most demanding steps in the CCSD and (T) algorithms scale as :math:`{\cal{O}}(o^2v^4)` and :math:`{\cal{O}}(o^3v^4)`, where :math:`o` and :math:`v` represent the number of oribitals that are occupied and unoccupied (virtual) in the reference function, respectively. By reducing the the size of the virtual space, the cost of evaluating these terms reduces by a factor of :math:`(v / v_{FNO})^4`, where :math:`v_{FNO}` represents the number of virtual orbitals retained after the FNO truncation. The general outline for the FNO procedure in |Psifour| is: (i) construct the virtual-virtual block of the unrelaxed MP2 one-particle density matrix (OPDM) (ii) diagonalize this block of the OPDM to obtain a set of natural virtual orbitals (iii) based on some occupancy threshold, determine which orbitals are unimportant and may be discarded (iv) project the virtual-virtual block of the Fock matrix onto the truncated space (v) construct semicanonical orbitals by diagonalizing the virtual-virtual block of the Fock matrix (vi) proceed with the QCISD(T) / CCSD(T) / MP4 computation in the reduced virtual space A second-order correction based upon the MP2 energies in the full and truncated spaces captures much of the missing correlation effects. More details on the implementation and numerical accuracy of FNO methods in |Psifour| can be found in Ref. [DePrince:2013:293]_\. FNO computations are controlled through the keywords |fnocc__nat_orbs| and |fnocc__occ_tolerance|, or by prepending a valid method name with "fno" in the energy call as :: energy('fno-ccsd(t)') If you wish to specify the number of active natural orbitals manually, use the keyword |fnocc__active_nat_orbs|. This keyword will override the keyword |fnocc__occ_tolerance|. QCISD(T), CCSD(T), MP4, and CEPA ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The FNOCC module in |Psifour| supports several related many-body quantum chemistry methods, including the CCSD(T) and QCISD(T) methods, several orders of many-body perturbation theory (MP2-MP4), and a family methods related to the coupled electron pair approximation (CEPA). Quadratic configuration interaction and coupled cluster ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The quadratic configuration interaction singles doubles (QCISD) method of Pople, Head-Gordon, and Raghavachari [Pople:1987:5968]_\ was originally presented as a size-consistent extension of configuration interaction singles doubles (CISD). The method can also be obtained as a simplified version of the coupled cluster singles doubles (CCSD) method [Purvis:1982]_\. Consider the set of equations defining CCSD: .. math:: :label: CCSD \langle \Psi_0 | (H - E) (1 + T_1 + T_2 + \frac{1}{2}T_1^2)|\Psi_0\rangle = 0, \\ \langle \Psi_i^a | (H - E) (1 + T_1 + T_2 + \frac{1}{2}T_1^2+T_1T_2+\frac{1}{3!}T_1^3)|\Psi_0\rangle = 0, \\ \langle \Psi_{ij}^{ab} | (H - E) (1 + T_1 + T_2 + \frac{1}{2}T_1^2 + T_1T_2+\frac{1}{3!}T_1^3+\frac{1}{2}T_2^2+\frac{1}{2}T_1^2T_2+\frac{1}{4!}T_1^4)|\Psi_0\rangle = 0, \\ where we have chosen the intermediate normalization, :math:`\langle \Psi_0| \Psi \rangle = 1`, and the symbols :math:`T_1` and :math:`T_2` represent single and double excitation operators. The QCISD equations can be obtained by omitting all but two terms that are nonlinear in :math:`T_1` and :math:`T_2`: .. math:: :label: QCISD \langle \Psi_0 | (H - E) (1 + T_1 + T_2)|\Psi_0\rangle = 0, \\ \langle \Psi_i^a | (H - E) (1 + T_1 + T_2 + T_1T_2)|\Psi_0\rangle = 0, \\ \langle \Psi_{ij}^{ab} | (H - E) (1 + T_1 + T_2 + \frac{1}{2}T_2^2)|\Psi_0\rangle = 0. \\ QCISD is slightly cheaper that CCSD computationally, but it retains the :math:`{\cal{O}}(o^2v^4)` complexity of the original equations. Just as in the familiar CCSD(T) method, the effects of connected triple excitations may be included noniteratively to yield the QCISD(T) method. Both the QCISD(T) and CCSD(T) methods are implemented for closed-shell references in the FNOCC module. .. _`sec:fnompn`: Many-body perturbation theory ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ QCI and CC methods are closely related to perturbation theory, and the MP2, MP3, and MP4(SDQ) correlation energies can be obtained as a free by-product of a CCSD or QCISD computation. The following is an example of the results for a computation run with the call ``energy('fno-qcisd')`` to :py:func:`~driver.energy`:: QCISD iterations converged! OS MP2 FNO correction: -0.000819116338 SS MP2 FNO correction: -0.000092278158 MP2 FNO correction: -0.000911394496 OS MP2 correlation energy: -0.166478414245 SS MP2 correlation energy: -0.056669079827 MP2 correlation energy: -0.223147494072 * MP2 total energy: -76.258836941658 OS MP2.5 correlation energy: -0.171225850256 SS MP2.5 correlation energy: -0.054028401038 MP2.5 correlation energy: -0.225254251294 * MP2.5 total energy: -76.260943698880 OS MP3 correlation energy: -0.175973286267 SS MP3 correlation energy: -0.051387722248 MP3 correlation energy: -0.227361008515 * MP3 total energy: -76.263050456101 OS MP4(SDQ) correlation energy: -0.180324322304 SS MP4(SDQ) correlation energy: -0.048798468084 MP4(SDQ) correlation energy: -0.230995119324 * MP4(SDQ) total energy: -76.266684566910 OS QCISD correlation energy: -0.181578117924 SS QCISD correlation energy: -0.049853548145 QCISD correlation energy: -0.231431666069 * QCISD total energy: -76.267121113654 The first set of energies printed corresponds to the second-order FNO correction mentioned previously. Results for many-body perturbation theory through partial fourth order are then provided. The notation MP4(SDQ) indicates that we have included all contributions to the correlation energy through fourth order, with the exception of that from connected triple excitations. One need not run a full QCISD or CCSD computation to obtain these perturbation theory results. The keywords for invoking perturbation theory computations are given below in Table :ref:`FNOCC Methods `. Full MP4 correlation energies are also available. .. _`sec:fnocepa`: Coupled electron pair approximation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Coupled-pair methods can be viewed as approximations to CCSD or as size-extensive modifications of CISD. The methods have the same complexity as CISD, and solving the CISD or coupled-pair equations requires fewer floating point operations than solving the CCSD. CISD, CCSD, and the coupled-pair methods discussed below all scale formally with the sixth power of system size, and, as with the QCISD method, CEPA methods retain :math:`{\cal{O}}(o^2v^4)` complexity of the CCSD equations. For a detailed discussion of the properties of various coupled-pair methods, see Ref. [Wennmohs:2008:217]_\. 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. 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` | +=========================+============================================================+==============================================+ | cisd | :math:`E_c` | :math:`E_c` | +-------------------------+------------------------------------------------------------+----------------------------------------------+ | 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` | +-------------------------+------------------------------------------------------------+----------------------------------------------+ .. comment | dci | :math:`E_c` | NA | .. comment +-------------------------+------------------------------------------------------------+----------------------------------------------+ 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. Note, however, that 1-electron properties are currently unavailable when coupling these stationary CEPA-like methods with frozen natural orbitals. Density-fitted coupled cluster ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Density fitting (DF) [or the resolution of the identity (RI)] and Cholesky decomposition (CD) techniques are popular in quantum chemistry to avoid the computation and storage of the 4-index electron repulsion integral (ERI) tensor and even to reduce the computational scaling of some terms. DF/CD-CCSD(T) computations are available in |Psifour|, with or without the use of FNOs, through the FNOCC module. The implementation and accuracy of the DF/CD-CCSD(T) method are described in Ref. [DePrince:2013:2687]_\. The DF-CCSD(T) procedure uses two auxiliary basis sets. The first set is that used in the SCF procedure, defined by the |scf__df_basis_scf| keyword. If this keyword is not specified, an appropriate -JKFIT set is automatically selected. This auxiliary set defines the ERI's used to build the Fock matrix used in the DF-CCSD(T) procedure. The second auxiliary set is used to approximate all other ERI's in the DF-CCSD(T) procedure. The choice of auxiliary basis is controlled by the keyword |fnocc__df_basis_cc|. By default, |fnocc__df_basis_cc| is the RI set (optimized for DFMP2) most appropriate for use with the primary basis. For example, if the primary basis is aug-cc-pVDZ, the default |fnocc__df_basis_cc| will be aug-cc-pVDZ-RI. Alternatively, the user can request that the DF-CCSD(T) procedure use a set of vectors defined by the Cholesky decomposition of the ERI tensor as the auxiliary basis. This feature is enabled by specifying |globals__cc_type| ``CD``. CD methods can be enabled in the SCF procedure as well, by specifying the |scf__scf_type| as ``CD``. The accuracy of the decomposition can be controlled through the keyword |fnocc__cholesky_tolerance|. .. comment This feature is enabled by specifying |fnocc__df_basis_cc| as "CHOLESKY". The following input file sets up a DF-CCSD(T) computation using CD integrals :: molecule h2o { 0 1 O H 1 1.0 H 1 1.0 2 104.5 } set { scf_type cd cc_type cd basis aug-cc-pvdz freeze_core true } energy('ccsd(t)') The resulting CCSD(T) correlation energy will be equivalent to that obtained from a conventional computation if |fnocc__cholesky_tolerance| is sufficiently small (e.g. :math:`10^{-9}`). .. _`sec:fnogn`: Gn theory ~~~~~~~~~ The FNOCC module contains all the components that comprise the Gn family of composite methods. Currently, only the G2 method is supported [Curtiss:1991:7221]_\. The G2 procedure may be called through the :py:func:`~driver.energy` wrapper: :: energy('gaussian-2') Supported methods ~~~~~~~~~~~~~~~~~ The various methods supported by the FNOCC module in |Psifour| are detailed in Table :ref:`FNOCC Methods `. Note that these methods are implemented for closed-shell references only. For open-shell references, the calls ``energy('mp2.5')``, ``energy('mp3')``, and ``energy('mp4')`` will default to implementations of these methods in other modules :ref:`Cross-module Redundancies `. .. _`table:fnocc_methods`: +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | name | calls method | type select | +=========================+=============================================================+===========================================+ | qcisd | quadratic configuration interaction singles doubles | |globals__ci_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | qcisd(t) | qcisd with perturbative triples | |globals__ci_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | mp2.5 | average of second- and third-order perturbation theories | |globals__mp_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | mp3 | third-order perturbation theory | |globals__mp_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | mp4(sdq) | fourth-order perturbation theory, minus triples contribution| |globals__mp_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | mp4 | full fourth-order perturbation theory | |globals__mp_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | lccd | linear ccd | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | cepa(0), lccsd | coupled electron pair approximation, variant 0 | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | cepa(1) | coupled electron pair approximation, variant 1 | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | cepa(3) | coupled electron pair approximation, variant 3 | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | acpf | averaged coupled-pair functional | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | aqcc | averaged quadratic coupled-cluster | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | cisd | configuration interaction with single and double excitations| |globals__ci_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-qcisd | qcisd with frozen natural orbitals | |globals__ci_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-qcisd(t) | qcisd(t) with frozen natural orbitals | |globals__ci_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-ccsd | coupled cluster singles doubles with frozen natural orbitals| |globals__cc_type| CONV, DF, CD | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-ccsd(t) | ccsd with perturbative triples and frozen natural orbitals | |globals__cc_type| CONV, DF, CD | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-mp3 | mp3 with frozen natural orbitals | |globals__mp_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-mp4(sdq) | mp4(sdq) with frozen natural orbitals | |globals__mp_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-mp4 | mp4 with frozen natural orbitals | |globals__mp_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-lccd | linear ccd with frozen natural orbitals | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-cepa(0), fno-lccsd | cepa(0) with frozen natural orbitals | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-cepa(1) | cepa(1) with frozen natural orbitals | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-cepa(3) | cepa(3) with frozen natural orbitals | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-acpf | acpf with frozen natural orbitals | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-aqcc | aqcc with frozen natural orbitals | |globals__cc_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ | fno-cisd | cisd with frozen natural orbitals | |globals__ci_type| CONV | +-------------------------+-------------------------------------------------------------+-------------------------------------------+ .. comment | df-ccsd | ccsd with density fitting | .. comment +-------------------------+-------------------------------------------------------------+ .. comment | df-ccsd(t) | ccsd(t) with density fitting | .. comment +-------------------------+-------------------------------------------------------------+ .. comment | fno-df-ccsd | ccsd with density fitting and frozen natural orbitals | .. comment +-------------------------+-------------------------------------------------------------+ .. comment | fno-df-ccsd(t) | ccsd(t) with density fitting and frozen natural orbitals | .. comment +-------------------------+-------------------------------------------------------------+ .. comment | dci | configuration interaction with double excitations | .. comment +-------------------------+-------------------------------------------------------------+ .. comment | fno-dci | dci with frozen natural orbitals | .. comment +-------------------------+-------------------------------------------------------------+ .. index:: FNOCC; basic-keywords Basic FNOCC Keywords ~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. include:: /autodir_options_c/mints__basis.rst .. include:: /autodir_options_c/globals__freeze_core.rst .. include:: /autodir_options_c/fnocc__r_convergence.rst .. include:: /autodir_options_c/fnocc__e_convergence.rst .. include:: /autodir_options_c/fnocc__maxiter.rst .. include:: /autodir_options_c/fnocc__diis_max_vecs.rst .. include:: /autodir_options_c/fnocc__nat_orbs.rst .. include:: /autodir_options_c/fnocc__occ_tolerance.rst .. include:: /autodir_options_c/fnocc__triples_low_memory.rst .. include:: /autodir_options_c/fnocc__cc_timings.rst .. include:: /autodir_options_c/fnocc__df_basis_cc.rst .. include:: /autodir_options_c/fnocc__cholesky_tolerance.rst .. include:: /autodir_options_c/fnocc__cepa_no_singles.rst .. include:: /autodir_options_c/fnocc__dipmom.rst .. index:: FNOCC; advanced-keywords Advanced FNOCC Keywords ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. include:: /autodir_options_c/fnocc__scs_mp2.rst .. include:: /autodir_options_c/fnocc__mp2_scale_os.rst .. include:: /autodir_options_c/fnocc__mp2_scale_ss.rst .. include:: /autodir_options_c/fnocc__scs_ccsd.rst .. include:: /autodir_options_c/fnocc__cc_scale_os.rst .. include:: /autodir_options_c/fnocc__cc_scale_ss.rst .. include:: /autodir_options_c/fnocc__run_mp2.rst .. include:: /autodir_options_c/fnocc__run_mp3.rst .. include:: /autodir_options_c/fnocc__run_mp4.rst .. include:: /autodir_options_c/fnocc__run_ccsd.rst .. include:: /autodir_options_c/fnocc__run_cepa.rst .. include:: /autodir_options_c/fnocc__compute_triples.rst .. include:: /autodir_options_c/fnocc__compute_mp4_triples.rst .. include:: /autodir_options_c/fnocc__dfcc.rst .. include:: /autodir_options_c/fnocc__cepa_level.rst