Slater-type Geminals Research Articles

The updates in Libcint 6: More integrals, API refinements, and SIMD optimization techniques.

Libcint is a library designed for the evaluation of analytical integrals for Gaussian type orbitals. It prioritizes simplicity, ease of use, and efficiency for the development of quantum chemistry programs. In the release of version 6.0, Libcint supports the computation of integrals for various operators, such as overlap, Coulomb, Gaunt, Breit, attenuated Coulomb, Slater-type geminals, and Yukawa potential, as well as arbitrary orders of derivatives for these operators. To enhance the usability of the library, Libcint provides a uniform function signature for all integral functions. A code generator is included to automate the implementation of new integrals. To achieve better performance on modern central processing unit architectures, the library employs explicit single instruction multiple data parallelization in the code implementation.

Comment on "Canonical transcorrelated theory with projected Slater-type geminals" [J. Chem. Phys. 136, 084107 (2012)

This Comment identifies errors in the formalism/implementation reported in Ref. 1 and discusses some aspects of that work that were not fully specified in the original publication. This Comment also provides reference numerical results for a simple system to ease future implementation of the approach by other researchers.

Doubly Occupied Pair Coupled Cluster F12 Approach.

Inspired by reports of the good performance of the doubly occupied pair coupled cluster (pCCD) theory in describing static electron correlation, we have introduced and implemented a variant thereof that includes single excitations and explicitly treats the dynamic electron correlation using the F12 methodology (pCCSD-F12). This drastically reduces the computation scaling with respect to the standard method using the full double-excitation operator (CCSD-F12). Slater-type geminals as a correlation factor, together with fixed cusp conditions, were used, which is known as the SP-ansatz. For sample model systems, we have investigated the performance of reference states constructed from either canonical or localized molecular orbitals. Finaly, the employment of Brueckner orbitals has been tested, which causes the single excitations to naturally vanish from the wave function expansion (B-pCCD-F12). Our test systems include different-sized rings of hydrogen atoms and dissociation curves for small molecules such as HF, N2, and CO2; and comparison with CCSD-F12 is presented for a series of reaction enthalpies.

Dispersion interactions in dimers of inert gases: Investigation by means of explicitly correlated coupled clusters CCSD(F12)(T)

Dispersion interactions in He2, Ne2, and Ar2 dimers are investigated using the explicitly correlated method CCSD(F12)(T) with numerical quadratures. The obtained energies of interaction agree well with the results from highly accurate CCSD(T) calculations described in works in which sets of bonded functions are used. It is shown that in order to attain high accuracy in estimating energies of interaction, the exponential parameter of Slater-type geminals in the CCSD(F12)(T) method must be optimized. The higher accuracy of the values of C 6 asymptotic coefficients found using the CCSD(F12)(T) method instead of the conventional CCSD(T) approach is established, demonstrating the importance of considering short-range electron correlation effects in asymptotic regions of potential surfaces where the interaction between atoms is of a long-range dispersion nature.

Explicitly correlated internally contracted multireference coupled-cluster singles and doubles theory: ic-MRCCSD(F12∗)

An explicitly correlated ansatz employing Slater-type geminals and cusp conditions is developed for the internally contracted multireference coupled-cluster singles and doubles method. Only the most important geminal terms are retained in the spirit of earlier work for single-reference theory. Throughout all our test calculations, the new ic-MRCCSD(F12∗) method improves the basis set convergence of many properties, e.g., spectroscopic constants or singlet–triplet splittings, with only little extra computational cost. If a perturbative correction for connected triples is included (the ic-MRCCSD(F12∗)+(T) method), very accurate results can be obtained even with minimal active spaces.

Canonical transcorrelated theory with projected Slater-type geminals

An effective Hamiltonian perturbed with explicit interelectronic correlation is derived from similarity transformation of Hamiltonian using a unitary operator with Slater-type geminals. The Slater-type geminal is projected onto the excitation (and deexcitation) component as in the F12 theory. Simplification is made by truncating higher-body operators, resulting in a correlated Hamiltonian which is Hermitian and has exactly the same complexity as the original Hamiltonian in the second quantized form. It can thus be easily combined with arbitrary correlation models proposed to date. The present approach constructs a singularity-free Hamiltonian a priori, similarly to the so-called transcorrelated theory, while the use of the canonical transformation assures that the effective Hamiltonian is two-body and Hermite. Our theory is naturally extensible to multireference calculations on the basis of the generalized normal ordering. The construction of the effective Hamiltonian is non-iterative. The numerical assessments demonstrate that the present scheme improves the basis set convergence of the post-mean-field calculations at a similar rate to the explicitly correlated methods proposed by others that couple geminals and conventional excitations.

Multireference F12 coupled cluster theory: The Brillouin-Wigner approach with single and double excitations

This Letter reports development and implementation of the explicitly correlated multireference Brillouin Wigner (MR BW-CC) coupled cluster method with Slater type geminals. The performance of the new approach is tested on the H4 model system and the dissociation curve of the fluorine molecule. Like in single reference methods, results show a dramatically improved convergence of total energies towards complete basis set limit as compared to a conventional MR BW-CC approach. In comparison with previously reported calculations with a linear correlation factor, there is a better performance for calculations in smaller basis sets.

Explicitly correlated coupled cluster methods with pair-specific geminals

Explicitly correlated MP2-F12 and CCSD(T)-F12 methods with orbital-pair-specific Slater-type geminals are proposed. The fixed amplitude ansatz of Ten-no is used, and different exponents of the Slater geminal functions can be chosen for core–core, core–valence, and valence–valence pairs. This takes care of the different sizes of the correlation hole and leads to improved results when inner-shell orbitals are correlated. The complications and the extra computational cost as compared to corresponding calculations with a single geminal are minor. The improved accuracy of the method is demonstrated for spectroscopic properties of Br2, As2, Ga2, Cu2, GaCl, CuCl, and CuBr, where the d-orbitals are treated as core.

Interference-corrected explicitly-correlated second-order perturbation theory

In the complete-basis-set (CBS) model chemistries, an interference correction is added to the total energy. It is added because the CBS-type extrapolation to the basis-set limit of second-order Møller–Plesset (MP2) perturbation theory usually overestimates the magnitude of the basis-set truncation error at higher levels of electron-correlation treatment. In this Letter, the question is pursued whether the CBS-type interference correction could also improve the computational results when the MP2 basis set limit is approached by means of explicitly-correlated MP2 theory using Slater-type geminals.

Communications: Explicitly correlated second-order Møller–Plesset perturbation method for extended systems

A formalism for the second-order Møller-Plesset perturbation method employing basis functions that depend explicitly on electron-electron distances (the MP2-R12 or F12 method) is derived and implemented into computer codes for extended systems periodic in one dimension. The excitation amplitudes on these functions are held fixed at values that satisfy the first-order cusp condition. Necessary many-electron integrals over Gaussian-type functions involving Slater-type geminals are evaluated by means of the resolution-of-the-identity approximation with a complementary auxiliary basis set. These integrals and thus the final correlation energy are shown to have the correct size dependence. The valence MP2 correlation energy of polyethylene near the complete basis-set limit is obtained and shown to be considerably greater in magnitude than the value obtained without the R12 treatment.

Analytic Calculation of First-order Molecular Properties at the Explicitly-correlated Second-order Møller-Plesset Level

Abstract Formulae are derived and implemented for the analytic calculation of first-order molecular properties at the level of explicitly-correlated second-order Møller-Plesset perturbation theory. In this theory, which is denoted as MP2-F12 theory, Slater-type geminals are used to expand the first-order wave function. A second-order perturbation theory correction for single excitations into a complementary auxliary basis set is also included. At the MP2-F12 level, it seems sufficient to restrict the analytic calculation of energy derivatives to the level of standard approximation A of MP2-F12 theory and to assume the extended Brillouin condition to hold. Smooth and rapid convergence towards the basis-set limit is observed for the dipole moments of a selection of small closed- and open-shell molecules when calculated at the RI-MP2-F12/2A*[T+V] + CABS singles level in augmented correlation-consistent polarized valence double-, triple-, and quadruple-zeta basis sets that have been optimized especially for use in MP2-F12 theory.

Accurate computational thermochemistry from explicitly correlated coupled-cluster theory

Explicitly correlated coupled-cluster theory has developed into a valuable computational tool for the calculation of electronic energies close to the limit of a complete basis set of atomic orbitals. In particular at the level of coupled-cluster theory with single and double excitations (CCSD), the space of double excitations is quickly extended towards a complete basis when Slater-type geminals are added to the wave function expansion. The purpose of the present article is to demonstrate the accuracy and efficiency that can be obtained in computational thermochemistry by a CCSD model that uses such Slater-type geminals. This model is denoted as CCSD(F12), where the acronym F12 highlights the fact that the Slater-type geminals are functions f(r 12) of the interelectronic distances r 12 in the system. The performance of explicitly correlated CCSD(F12) coupled-cluster theory is demonstrated by computing the atomization energies of 73 molecules (containing H, C, N, O, and F) with an estimated root-mean-square deviation from the values compiled in the Active Thermochemical Tables of σ = 0.10 kJ/mol per valence electron. To reach this accuracy, not only the frozen-core CCSD basis-set limit but also high-order excitations (connected triple and quadruple excitations), core–valence correlation effects, anharmonic vibrational zero-point energies, and scalar and spin–orbit relativistic effects must be taken into account.

Atomization energies from coupled-cluster calculations augmented with explicitly-correlated perturbation theory

The atomization energies of the 105 molecules in the test set of Bakowies [D. Bakowies, J. Chem. Phys. 127 (2007) 084105] have been computed with an estimated standard deviation (from the values compiled in the Active Thermochemical Tables) of ±0.1kJ/mol per valence electron in the molecule. Equilibrium geometries and harmonic vibrational frequencies were calculated at the all-electron CCSD(T)/cc-pCVTZ level, that is, at the level of coupled-cluster theory with singles, doubles and non-iterative triples in a correlation-consistent polarized core–valence triple-zeta basis. Single-point energy calculations were performed at the all-electron CCSD(T) level in a correlation-consistent polarized core–valence quadruple-zeta basis (cc-pCVQZ), and several corrections were added: (i) a correction for the basis-set truncation error, obtained from second-order perturbation theory using Slater-type geminals (MP2-F12 theory), (ii) a correction for the effect of anharmonicity on the zero-point vibrational energy, (iii) a relativistic correction, (iv) a correction for the difference between the full CCSDT model (coupled-cluster theory with singles, doubles and triples) and the CCSD(T) approximation, and (v) a correction for connected quadruple excitations obtained from CCSDT(Q) calculations. The correction for the basis-set truncation error was obtained from MP2-F12 calculations by scaling the MP2 basis-set truncation error by an empirically optimized “interference factor” of fint=0.78. The reference values from the Active Thermochemical Tables for 73 molecules in the test set, the equilibrium geometries, the harmonic vibrational frequencies, and all of the energy corrections represent valuable data for performance assessments of additivity schemes that will be developed in the future, in which the basis-set truncation error will be calculated at the level of coupled-cluster theory using Slater-type geminals (CC-F12 theory). Such a scheme will be free of empirical corrections and scaling factors.

Explicitly correlated second-order perturbation theory calculations on molecules containing heavy main-group elements

Slater-type geminals (STGs) have been used as explicitly correlated two-electron basis functions for calculations on the hydrides of N–As and Sb (as well as on the hydrides of O–Se and F–Br with similar, not reported results) in various one-electron basis sets of Gaussian atomic orbitals. The performance of the explicitly correlated theory has been assessed with respect to the exponent of the STG, for example, by using different exponents for individual pair correlation functions and pair energies. It is shown that a correlation factor with an exponent of \({\gamma = 1.4 a_{0}^{-1}}\) can give reliable results within 1% from the basis-set limit for all investigated molecules in an aug-cc-pVQZ basis set for the valence shells, using fixed amplitudes for the STGs in a diagonal orbital-invariant formulation of the theory. The use of relativistic effective core potentials (RECPs) in explicitly correlated second-order perturbation theory has been investigated.

Slater-type geminals in explicitly-correlated perturbation theory: application to n-alkanols and analysis of errors and basis-set requirements

In the recent years, Slater-type geminals (STGs) have been used with great success to expand the first-order wave function in an explicitly-correlated perturbation theory. The present work reports on this theory's implementation in the framework of the Turbomole suite of programs. A formalism is presented for evaluating all of the necessary molecular two-electron integrals by means of the Obara-Saika recurrence relations, which can be applied when the STG is expressed as a linear combination of a small number (n) of Gaussians (STG-nG geminal basis). In the Turbomole implementation of the theory, density fitting is employed and a complementary auxiliary basis set (CABS) is used for the resolution-of-the-identity (RI) approximation of explicitly-correlated theory. By virtue of this RI approximation, the calculation of molecular three- and four-electron integrals is avoided. An approximation is invoked to avoid the two-electron integrals over the commutator between the operators of kinetic energy and the STG. This approximation consists of computing commutators between matrices in place of operators. Integrals over commutators between operators would have occurred if the theory had been formulated and implemented as proposed originally. The new implementation in Turbomole was tested by performing a series of calculations on rotational conformers of the alkanols n-propanol through n-pentanol. Basis-set requirements concerning the orbital basis, the auxiliary basis set for density fitting and the CABS were investigated. Furthermore, various (constrained) optimizations of the amplitudes of the explicitly-correlated double excitations were studied. These amplitudes can be optimized in orbital-variant and orbital-invariant manners, or they can be kept fixed at the values governed by the rational generator approach, that is, by the electron cusp conditions. Electron-correlation effects beyond the level of second-order perturbation theory were accounted for by conventional coupled-cluster calculations with single, double and perturbative triple excitations [CCSD(T)]. The explicitly-correlated perturbation theory results were combined with CCSD(T) results and compared with literature data obtained by basis-set extrapolation.

Initiation of explicitly correlated Slater-type geminal theory

We employ the Slater-type function as a geminal basis function to incorporate the inter-electron distance in explicitly correlated theory. It is shown that the use of the Slater-type geminals confers numerical and computational advantages over the previous explicitly correlated methods. The performance of the resulting method is examined in some benchmark calculations at the second order Møller–Plesset perturbation theory. The results reveal that the Slater-type function is promising compared to the ordinary Gaussian-type geminals and linear r 12 function.

Nuclear fusion rates of muonic molecular ions.

Using basis sets of random-tempered Slater-type geminals, we have calculated the nuclear fusion rates for the J=0,1 bound states of the mesomolecular ions xyμ(x,y=p,d,t).Received 25 September 1989DOI:https://doi.org/10.1103/PhysRevA.41.2854©1990 American Physical Society

High-accuracy calculation of muonic molecules using random-tempered basis sets.

We use random-tempering formulas and explicitly correlated Slater-type geminals to calculate the S and P bound-state energies of the muonic molecules xy\ensuremath{\mu}, where x,y=p,d,t. The final binding energies are accurate to about 1 \ensuremath{\mu}eV or better except for the weakly bound td\ensuremath{\mu}(11) state. For this state we get a binding energy of 0.660 1721 eV, which is better than any previous calculation.