Density second-order correction

This State for optimization and/or second-order correction Total Energy, E(Cis) = -77.8969983928 "Copying the Cisingles density for this state as the 1-particle RhoCI density. 

A Perturbation Theory is developed for treating a system of n electrons in which the Hartree-Fock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy and the charge density of the system is zero. The expression for the second order correction for the energy greatly simplifies because of the special property of the zero order solution. It is pointed out that the development of the higher order approximation involves only calculations based on a definite one-body problem. 

P/h can be interpreted as an effective spin density of this open shell system. Similarly to the electron binding exjvession there is no first order contribution in the correlation potential, that is, = 0, so that 5 is correct through second order. However, the second order correction in the electron correction for... 

Terms containing the W intermediates no longer contain a factor of The energy-independent, third-order term, Epp (oo), is a Coulomb-exchange matrix element determined by second-order corrections to the density matrix, where... 

Assuming that an ab initio or semiempirical technique has been used to obtain p(r), we address the important question of how the calculated electrostatic potential depends on the nature of the wavefunction used for computing p(r). Historically, and today as well, most ab initio calculations of V(r) for reasonably sized molecules have been based on self-consistent-field (SCF) or near Hartree-Fock wavefunctions and therefore do not reflect electron correlation. Whereas the availability of supercomputers has made post-Hartree-Fock calculations of V(r) (which include electron correlation) a realistic possibility even for molecules with 5 to 10 first-row atoms, there is reason to believe that such computational levels are usually not necessary and not warranted. The M0l er-Plesset theorem states that properties computed from Hartree-Fock wavefunctions using one-electron operators, as is V(r), are correct through first order " any errors are no more than second-order effects. Whereas second-order corrections may not always be insignificant, several studies have shown that near-Hartree-Fock electron densities are affected to only a minor extent by the inclusion of correlation.The limited evidence available suggests that the same is true of V(r), ° ° as is indicated also by the following example. 

The alternating direction implicit method (i.e., ADI) is employed to calculate transverse diffusion in the x direction via second-order-correct finite differences for a second derivative using unknown molar densities at Zk+i-Hence,... 

However, this introduces another unknown molar density at x i which arises from the second-order correct central difference expression for d CAl x atxo, yj. The boundary condition at the symmetry plane is used to relate Ca at x i to Ca at xi via a second-order correct central difference expression for a first derivative... 

One more algebraic eqnation is required to solve for all unknown molar densities at Zk+i It is not advantageons to write the mass balance at the catalytic surface (i.e., at xatx-i-i) because the no-slip boundary condition at the wall stipnlates that convective transport is identically zero. Hence, one relies on the radiation boundary condition to generate eqnation (23-46). Diffusional flux of reactants toward the catalytic snrface, evalnated at the surface, is written in terms of a backward difference expression for a first-derivative that is second-order correct, via equation (23-40). This is illnstrated below at Xwaii = x x+i for equispaced data ... 

For slowly varying densities, the kinetic energy functional can be represented by one of its gradient expansions. The gradient expansion of the kinetic energy density is not unique since it relies upon different derivations techniques [35], which yield or not a contibution of the laplacian of the density in the second order correction. In the following we will consider the expansion expression which does not involve V /i(r) ... 

This procedure can be then iterated by taking further derivative of Eq. (4.465) with respect to the density, solving the obtained equation until the second order correction over above first order solution (4.469),... 

Fermi functional of Equation 5.4, and the last term gives the second-order correction of Equation 5.23. On the other hand, for excited states, the second term cannot be written in terms of the density. Furthermore, in many cases, the last term increases exponentially in regions asymptotically far from the system, as discussed below. 

Similarly, the MP second-order correction to the electron density can be defined... 

This is often called the unrelaxed second-order correction to the density matrix in order to distinguish it from the relaxed density matrix, which will be... 

Compared to the A-matrix in RPA, Eq. (10.18), one obtains in second order two additional contributions, which consist of contractions of two-electron repulsion integrals with the first-order doubles correlation coefficients defined in Eq. (9.67), and one term that contains the second-order correction to the density matrix, Eqs. (9.116) and (9.117). The latter contribution... 

The second-order corrections to the h part of the property gradient imply that a SOPPA first-order density matrix will not be contracted with property integrals in the molecular orbital basis. However, defining a kind of MP2 correction to the molecular orbitals as... 

This keeps essentially the structure of the SOPPA equations but replaces in all matrix elements the first-order MP doubles correlation coefficients, Eq. (9.67), and the second-order MP singles correlation coefficient, Eq. (9.71), by coupled cluster singles and doubles amplitudes. In the earlier coupled cluster singles and doubles polarization propagator approximation (CCSDPPA) (Geertsen et al., 1991a), a precursor to SOPPA(CCSD), this was done only partially and in particular not in the second-order correction to the density matrix Very recently, a third method (Kjaer... 

In Table 13.1 some results for the electric dipole moment (Packer et al., 1994) of the hydrogen halides, HX, and methyl halides, CH3X, are shown. They are calculated with the SCF density matrix, Eq. (9.112),with the unrelaxed second-order (MP2) density matrix in Eqs. (9.116)-(9.118) and with the relaxed second-order (MP2) density matrix in Eq. (12.5). The results for the dipole moments are clearly improved by the second-order correction to the MP density matrix. However, no clear trend is observable for the comparison of the relaxed and unrelaxed MP2 density matrix. Correlation at this level reduces the dipole moments on average by 9%. The root-mean-square percentage deviation of the unrelaxed MP2 results from the experimental equilibrium geometry values is 3.6% with a maximum and minimum deviation of 5.0% and -1.4%, respectively. 

S. Grimme,/. Chem. Phys., 124, 034108 1-16 (2006). Semiempirical Hybrid Density Functional with Perturbative Second-Order Correction. 

Hudson also introduced a second-order correction. Mavko et al. (1998) note The second-order expansion is not a uniformly converging series and predicts increasing moduli with crack density beyond the formal limit. Better results will be obtained by using just the first-order correction rather than inappropriately using the second-order correction . 

There is a rather wide disparity between experiment and the second-order cumulate prediction for small T, equation 7.123 correctly predicts both the values of equilibrium density for large T and the fact that the system undergoes a sharp... 

Here the matrix V contains the effect of the nuclear displacements therefore the inhomogeneous first term to the right is a driving term the second term to the right is of second order in the driving effect, and could be dropped in calculations. Formally, the solution for the configuration density matrix correction is... 

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