Function, variational

Standard deviation multiplier, Standard Normal variate Function of... 

The energy of the bond for this function too is essentially resonance energy. Dickinson2 introduced an additional term, dependent on two additional parameters, in order to take polarization into account. He wrote for the (not yet normalized) variation function... 

The van der Waals interaction energy of two hydrogen atoms at large intemuclear distances is discussed by the use of a linear variation function. By including in the variation function, in addition to the unperturbed wave function, 26 terms for the dipole-dipole interaction, 17 for the dipole-quadrupole interaction, and 26 for the quadrupole-quadrupole interaction, the... 

It is interesting to note that the value A = 6.4976 found by Hasse for the dipole-dipole coefficient by the use of a variation function (Vi) oo (V2) j 1 +11 (A +Br1r2+Cri2r +Dr1 ,r/) J is very close to our value for pi—4, p2—4, which is based on a variation function involving all terms (unsymmetric as well as symmetric) out to rfr . This indicates that the unsymmetric terms are of minor importance. 

The precedings sections concentrated on single determinant variational functions. One may wonder whether going to multieonfigurational SCF funetions will restore symmetry and when. 

It should be noted that the integral equations [2] determining the elements Kp B. aE derived as an energy-variational problem, correspond also to the stationary condition of the variational functional proposed by Newton (9). Thus the K-matrix elements obeying equation [2] guarantee a stationary value for the K-matrix on the energy shell. 

A convenient and widely used form for the trial function 0 is the linear variation function... 

The numerical value of S is listed in Table 9.1. The simple variation function (9.88) gives an upper bound to the energy with a 1.9% error in comparison with the exact value. Thus, the variation theorem leads to a more accurate result than the perturbation treatment. Moreover, a more complex trial function with more parameters should be expected to give an even more accurate estimate. 

Levy, M., 1979, Universal Variational Functionals of Electron Densities, First Order Density Matrices, and Natural Spin Orbitals and Solution of the v-Representability Problem , Proc. Natl. Acad. Sci. USA, 16, 6062. 

As a simple proof of the variational theorem, consider the case in which 4> tJ/0. The variational function can be expanded in terms of die complete... 

It is often convenient to employ a variation function that it is a linear combination of suitably chosen functions x thus,... 

The use of a linear variation function was summarized in the previous section. For the example of a diatomic molecule the set of simultaneous eauations fEa. (125VI becomes... 

Although, in most cases it may not be possible to make as fortunate a choice of variation function as in the previous example, even poor approximations to the actual wave function may yield surprisingly close approximations of the energy. 

The secular equation that corresponds to this linear variational function... 

In a well known practical but approximate method to solve the GS problem, known as the Hartree-Fock (HF) approximation (see e.g. [10]), the domain of variational functions P in Eq. (9) is narrowed to those that are a single Slater determinant (D) 9 d, constructed out of orthonormal spin orbitals tj/iix) ... 

Here, by definition, there is no electron-electron interaction operator while iTj is defined similarly to tT in Eq. (12), but with vir) replaced by tis( ). The variational function can be chosen to be determinantal, because such a function describes a noninteracting system. By performing the minimization in two steps external over n and internal over -> n [compare Eqs. (13)-(16)] we arrive at... 

The first inequality follows immediately from minimization (102), while the second one holds due to the fact that the variational functional space involved in minimization (97) is narrower than in (25). 

In 1979, an elegant proof of the existence was provided by Levy [10]. He demonstrated that the universal variational functional for the electron-electron repulsion energy of an A -representable trial 1-RDM can be obtained by searching all antisymmetric wavefunctions that yield a fixed D. It was shown that the functional does not require that a trial function for a variational calculation be associated with a ground state of some external potential. Thus the v-representability is not required, only Al-representability. As a result, the 1-RDM functional theories of preceding works were unified. A year later, Valone [19] extended Levy s pure-state constrained search to include all ensemble representable 1-RDMs. He demonstrated that no new constraints are needed in the occupation-number variation of the energy functional. Diverse con-strained-search density functionals by Lieb [20, 21] also afforded insight into this issue. He proved independently that the constrained minimizations exist. 

M. Levy, Universal variational functionals of electron-densities, Ist-order density-matrices, and natural spin-orbitals and solution of the v-representability problem. Pmc. Natl. Acad. Sci. U.S.A. 76(12), 6062-6065 (1979). 

Equation (2.18) is a linear variation function. (The summation indices prevent double-counting of excited configurations.) The expansion coefficients cq, c, c%, and so on are varied to minimize the variational integral. o) is a better approximation than l o)- In principle, if the basis were complete. Cl would provide an exact solution. Here we use a truncated expansion retaining only determinants D that differ from I Tq) by at most two spin orbitals this is a singly-doubly excited Cl (SDCI). 

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