Slater-type correlation function

Another critical advance responsible for the success of MP2-R12 is the introduction of nonlinear correlation functions, in particular, the Slater-type correlation function, 1— exp(—yr 2), of Ten-no [29]. It is asymptotically linear in ru near the coalescence point and, hence, satisfies the cusp condition in the leading order. Unlike the linear correlation function, the Slater-type correlation function... 

Figures 2 and 3 plot the valence correlation energies of Ne and FH obtained by various combinations of the CC or CC-R12 methods (using Ten-no s Slater-type correlation function) and basis sets (see ref. 35 for details). These figures are the stunning illustration of the extremely rapid convergence of correlation energies...

In Equations 10-12 A is the antisymmetrizer Xi(r3) ia a real-valued Slater type basis function (1)2) is an approximate target ground state (real-valued) wavefunction and jj(l,2,3) is a real-valued correlating configuration The three parameters k, and e were taken as nonlinear variational parameters and were varied to (approximately) satisfy the conditions... 

These equations are expressed in the spin-orbital formalism and the products of orbitals are assumed to be antisymmetrized. The coefficients are the explicitly correlated analogues of the conventional amplitudes. The xy indices refer to the space of geminal replacements which is usually spanned by the occupied orbitals. The operator Q12 in Eq. (21) is the strong orthogonality projector and /12 is the correlation factor. In Eq. (18) the /12 correlation factor was chosen as linear ri2 term. It is not necessary to use it in such form. Recent advances in R12 theory have shown that Slater-type correlation factors, referred here as /12, are advantageous. Depending on the choice of the Ansatz of the wave function, the formula for the projector varies, but the detailed discussion of these issues is postponed until Subsection 4.2. The minimization of the Hylleraas functional... 

The presence of the projector Q12 assures that the space spanned by the conventional double excitations is strongly orthogonal on the occupied space. Within the current work two Ansatze for Q12 are considered. To be consistent with the literature we will refer to Ansatz 1 and Ansatz 3 (abbreviated as Anl and An3, respectively). The /12 term is an arbitrary function of the distance between the electrons ri2. Within the current work the linear ri2 and Slater type correlation factors exp(—7ri2) are considered. Due to the computational convenience, the latter one was implemented as a linear combination of Gaussian functions (LCG) in the following way... 

All explicitly correlated calculations were performed at the CCSD(F12) level of theory, as implemented in the TurbomOLE program [58, 69]. The Slater-type correlation factor was used with the exponent 7 = 1.0 aQ. It was approximated by a linear combination of six Gaussian functions with linear and nonlinear coefficients taken from Ref. [44]. The CCSD(F12) electronic energies were computed in an all-electron calculation with the d-aug-cc-pwCV5Z basis set [97]. For all cases we used full CCSD(F12) model (see Subsection 4.9 for the discussion about models implemented in Turbomole), the open-shell species were computed with a UHF reference wave function. The explicitly correlated contributions to the relative quantities are collected in Tables 10 and 11 under the label F12 . 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Quantum mechanics (QM) can be further divided into ab initio and semiempiri-cal methods. The ab initio approach uses the Schrodinger equation as the starting point with post-perturbation calculation to solve electron correlation. Various approximations are made that the wave function can be described by some functional form. The functions used most often are a linear combination of Slater-type orbitals (STO), exp (-ax), or Gaussian-type orbitals (GTO), exp (-ax2). In general, ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Self-consistency is achieved by a procedure in which a set of orbitals is assumed and the electron-electron repulsion is calculated. This energy is then used to calculate a new set of orbitals, and these in turn are used to calculate a new repulsion energy. The process is continued until convergence occurs and self-consistency is achieved. 

The one-electron Kohn-Sham equations were solved using the Vosko-Wilk-Nusair (VWN) functional [27] to obtain the local potential. Gradient correlations for the exchange (Becke fimctional) [28] and correlation (Perdew functional) [29] energy terms were included self-consistently. ADF represents molecular orbitals as linear combinations of Slater-type atomic orbitals. The double- basis set was employed and all calculations were spin unrestricted. Integration accuracies of 10 -10 and 10 were used during the single-point and vibrational frequency calculations, respectively. The cluster size chosen for Ag or any bimetallic was... 

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