Toru Shiozaki

Explicitly correlated multireference configuration interaction: MRCI-F12

An internally contracted multireference configuration interaction is developed which employs wave functions that explicitly depend on the electron-electron distance (MRCI-F12). This MRCI-F12 method has the same applicability as the MRCI method, while having much improved basis-set convergence with little extra computational cost. The F12b approximation is used to arrive at a computationally efficient implementation. The MRCI-F12 method is applied to the singlet-triplet separation of methylene, the dissociation energy of ozone, properties of diatomic molecules, and the reaction barrier and exothermicity of the F + H(2) reaction. These examples demonstrate that already with basis sets of moderate size the method provides near complete basis set MRCI accuracy, and hence quantitative agreement with the experimental data. As a side product, we have also implemented the explicitly correlated multireference averaged coupled pair functional method (MRACPF-F12).

Communication: Extended multi-state complete active space second-order perturbation theory: Energy and nuclear gradients

The extended multireference quasi-degenerate perturbation theory, proposed by Granovsky [J. Chem. Phys. 134, 214113 (2011)], is combined with internally contracted multi-state complete active space second-order perturbation theory (XMS-CASPT2). The first-order wavefunction is expanded in terms of the union of internally contracted basis functions generated from all the reference functions, which guarantees invariance of the theory with respect to unitary rotations of the reference functions. The method yields improved potentials in the vicinity of avoided crossings and conical intersections. The theory for computing nuclear energy gradients for MS-CASPT2 and XMS-CASPT2 is also presented and the first implementation of these gradient methods is reported. A number of illustrative applications of the new methods are presented.

Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12

An explicitly correlated complete active space second-order perturbation (CASPT2-F12) method is presented which strongly accelerates the convergence of CASPT2 energies and properties with respect to the basis set size. A Slater-type geminal function is employed as a correlation factor to represent the electron-electron cusp of the wave function. The explicitly correlated terms in the wave function are internally contracted. The required density matrix elements and coupling coefficients are the same as in conventional CASPT2, and the additional computational effort for the F12 correction is small. The CASPT2-F12 method is applied to the singlet-triplet splitting of methylene, the dissociation energy of ozone, and low-lying excited states of pyrrole.

Explicitly correlated coupled-cluster singles and doubles method based on complete diagrammatic equations.

The explicitly correlated coupled-cluster singles and doubles (CCSD-R12) and related methods-its linearized approximation CCSD(R12) and explicitly correlated second-order Moller-Plesset perturbation method-have been implemented into efficient computer codes that take into account point-group symmetry. The implementation has been largely automated by the computerized symbolic algebra SMITH that can handle complex index permutation symmetry of intermediate tensors that occur in the explicitly correlated methods. Unlike prior implementations that invoke the standard approximation or the generalized or extended Brillouin condition, our CCSD-R12 implementation is based on the nontruncated formalisms [T. Shiozaki et al., Phys. Chem. Chem. Phys. 10, 3358 (2008)] in which every diagrammatic term that arises from the modified Ansatz 2 is evaluated either analytically or by the resolution-of-the-identity insertion with the complementary auxiliary basis set. The CCSD-R12 correlation energies presented here for selected systems using the Slater-type correlation function can, therefore, serve as benchmarks for rigorous assessment of other approximate CC-R12 methods. Two recently introduced methods, CCSD(R12) and CCSD(2)(R12), are shown to be remarkably accurate approximations to CCSD-R12.

Equations of explicitly-correlated coupled-cluster methods

The tensor contraction expressions defining a variety of high-rank coupled-cluster energies and wave functions that include the interelectronic distances (r(12)) explicitly (CC-R12) have been derived with the aid of a newly-developed computerized symbolic algebra smith. Efficient computational sequences to perform these tensor contractions have also been suggested, defining intermediate tensors-some reusable-as a sum of binary tensor contractions. smith can elucidate the index permutation symmetry of intermediate tensors that arise from a Slater-determinant expectation value of any number of excitation, deexcitation and other general second-quantized operators. smith also automates additional algebraic transformation steps specific to R12 methods, i.e. the identification and isolation of the special intermediates that need to be evaluated analytically and the resolution-of-the-identity insertion to facilitate high-dimensional molecular integral computation. The tensor contraction expressions defining the CC-R12 methods including through the connected quadruple excitation operator (CCSDTQ-R12) have been documented and efficient computational sequences have been suggested not just for the ground state but also for excited states via the equation-of-motion formalism (EOM-CC-R12) and for the so-called Lambda equation (Lambda-CC-R12) of the CC analytical gradient theory. Additional equations (the geminal amplitude equation) arise in CC-R12 that need to be solved to determine the coefficients multiplying the r(12)-dependent factors. The operation cost of solving the geminal amplitude equations of rank-k CC-R12 and EOM-CC-R12 (right-hand side) scales as O(n(6)) (k = 2) or O(n(7)) (k > or = 3) with the number of orbitals n and is surpassed by the cost of solving the usual amplitude equations O(n(2k+2)). While the complexity of the geminal amplitude equations of Lambda- and EOM-CC-R12 (left-hand side) nominally scales as O(n(2k+2)), it is less than that of the other O(n(2k+2)) terms in the usual amplitude equations. This suggests that the unabridged equations should be solved in high-rank CC-R12 for benchmark accuracy.

Higher-order explicitly correlated coupled-cluster methods.

Efficient computer codes for the explicitly correlated coupled-cluster (CC-R12 or F12) methods with up to triple (CCSDT-R12) and quadruple excitations (CCSDTQ-R12), which take account of the spin, Abelian point-group, and index-permutation symmetries and are based on complete diagrammatic equations, have been implemented with the aid of the computerized symbolic algebra SMITH. Together with the explicitly correlated coupled-cluster singles and doubles (CCSD-R12) method reported earlier [T. Shiozaki et al., J. Chem. Phys. 129, 071101 (2008)], they form a hierarchy of systematic approximations (CCSD-R12<CCSDT-R12<CCSDTQ-R12) that converge very rapidly toward the exact solutions of the polyatomic Schrodinger equations with respect to both the highest excitation rank and basis-set size. Using the Slater-type function exp(-gamma r(12)) as a correlation function, a CC-R12 method can provide the aug-cc-pV5Z-quality results of the conventional CC method of the same excitation rank using only the aug-cc-pVTZ basis set. Combining these CC-R12 methods with the grid-based, numerical Hartree-Fock equation solver [T. Shiozaki and S. Hirata, Phys. Rev. A 76, 040503(R) (2007)], the solutions (eigenvalues) of the Schrodinger equations of neon, boron hydride, hydrogen fluoride, and water at their equilibrium geometries have been obtained as -128.9377+/-0.0004, -25.2892+/-0.0002, -100.459+/-0.001, and -76.437+/-0.003 E(h), respectively, without resorting to complete-basis-set extrapolations. These absolute total energies or the corresponding correlation energies agree within the quoted uncertainty with the accurate, nonrelativistic, Born-Oppenheimer values derived experimentally and/or computationally.

Multireference explicitly correlated F12 theories

We review our recent developments in multireference explicitly correlated F12 theories (explicitly correlated internally contracted multireference perturbation and multireference configuration interaction theories) that achieve near-basis-set-limit accuracy of the underlying multireference electron correlation methods with basis sets of medium size. The applicability of the multireference F12 theories is the same as that of their non-F12 counterpart, and therefore it is a computational tool with predictive accuracy for complicated electronic structures with strong correlation. A comparison with the earlier developments by others is also discussed.

Explicitly correlated multireference configuration interaction with multiple reference functions: Avoided crossings and conical intersections

We develop an explicitly correlated multireference configuration interaction method (MRCI-F12) with multiple reference functions. It can be routinely applied to nearly degenerate molecular electronic structures near conical intersections and avoided crossings, where the reference functions are strongly mixed in the correlated wave function. This work is a generalization of the MRCI-F12 method for electronic ground states, reported earlier by Shiozaki et al. [J. Chem. Phys. 134, 034113 (2011)]. The so-called F12b approximation is used to arrive at computationally efficient formulas. The doubly external part of the wave function is expanded in terms of internally contracted configurations generated from all the reference functions. In addition, we introduce a singles correction to the CASSCF reference energies, which is applicable to multi-state calculations. As examples, we present numerical results for the avoided crossing of LiF, excited states of ozone, and the H(2) + OH (A(2)Σ(+)) reaction.

Second- and third-order triples and quadruples corrections to coupled-cluster singles and doubles in the ground and excited states

Second- and third-order perturbation corrections to equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) incorporating excited configurations in the space of triples [EOM-CCSD(2)T and (3)T] or in the space of triples and quadruples [EOM-CCSD(2)TQ] have been implemented. Their ground-state counterparts--third-order corrections to coupled-cluster singles and doubles (CCSD) in the space of triples [CCSD(3)T] or in the space of triples and quadruples [CCSD(3)TQ]--have also been implemented and assessed. It has been shown that a straightforward application of the Rayleigh-Schrodinger perturbation theory leads to perturbation corrections to total energies of excited states that lack the correct size dependence. Approximations have been introduced to the perturbation corrections to arrive at EOM-CCSD(2)T, (3)T, and (2)TQ that provide size-intensive excitation energies at a noniterative O(n(7)), O(n(8)), and O(n(9)) cost (n is the number of orbitals) and CCSD(3)T and (3)TQ size-extensive total energies at a noniterative O(n(8)) and O(n(10)) cost. All the implementations are parallel executable, applicable to open and closed shells, and take into account spin and real Abelian point-group symmetries. For excited states, they form a systematically more accurate series, CCSD1 eV) and the ground-state wave function has single-determinant character. In other cases, however, the corrections tend to overestimate the triples and quadruples effects, the origin of which is discussed. For ground states, the third-order corrections lead to a rather small improvement over the highly effective second-order corrections [CCSD(2)T and (2)TQ], which is a manifestation of the staircase convergence of perturbation series.

Explicitly correlated combined coupled-cluster and perturbation methods

Coupled-cluster singles and doubles (CCSD) or coupled-cluster singles, doubles, and triples (CCSDT) with noniterative, perturbation corrections for higher-order excitations have been extended to include the basis functions that explicitly depend on interelectronic distances (r(12)) in the wave function expansions with the aim of dramatically accelerating the basis-set convergence of correlation energies. The extension has been based on the so-called R12 (or F12) scheme and applied to a second-order triples correction to CCSD [CCSD(2)(T)-R12], a second-order triples and quadruples correction to CCSD [CCSD(2)(TQ)-R12], a third-order triples correction to CCSD [CCSD(3)(T)-R12], and a second-order quadruples correction to CCSDT [CCSDT(2)(Q)-R12]. A simplified R12 treatment suggested by Fliegl et al. [J. Chem. Phys. 122, 084107 (2005)] has been combined with some of these methods, introducing CCSD(2)(T)(R12) and CCSD(2)(TQ)(R12). The CCSD(T)-R12 method has also been developed as an approximation to CCSD(2)(T)-R12. These methods have been applied to dissociation of hydrogen fluoride and double dissociation of water. For the molecules at their equilibrium geometries, molecular properties predicted by these methods converge extremely rapidly toward the complete-correlation, complete-basis-set limits with respect to the cluster excitation rank, perturbation order, and basis-set size. Although the R12 scheme employed in this work does not improve the basis-set convergence of connected triples or quadruples corrections, the basis-set truncation errors in these contributions have roughly the same magnitude as small residual basis-set truncation errors in the connected singles and doubles contributions even in the dissociation of hydrogen fluoride. In the double dissociation of water, the basis-set truncation errors in the connected triples contribution can be a few times as great as those in the connected singles and doubles contributions.