Variation function linear

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... 

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

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... 

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

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). 

In the last section, our calculation used only the function of Eq. (2.9), what is now called the covalent bonding function. According to our discussion of linear variation functions, we should see an improvement in the energy if we perform a two-state calculation that also includes the ionic function. 

The perceptive reader may already have observed that the functions we use can take many forms. Consider the non-Hermitian idempotent f/g)VJf. Using the permutations interconverting standard tableaux, one finds that (f/g) PMitj S i = 1,...,/ is a set of linearly independent functions (if S has no double occupancy). Defining a linear variation function in terms of these. 

A special type of variational function is the linear variation function. Here,

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Fio. 26-1.—Figure showing the interleaving of energy values for linear variation functions with added terms. 

Each of the functions described in Table 46-2 is an approximate solution of the wave equation for three hydrogen atoms it is therefore reasonable to consider the sum of them with undetermined coefficients as a linear variation function. The determination of the coefficients and the energy values then requires the solution of a secular equation (Sec. 26d) of eight rows and columns, a typical element of which is... 

The application of Eq. (81) in the method of linear variation functions is important for the later discussion in Sections III-A and III-B. Suppose 9) is a sum of linearly independent functions coefficients multiplying each of these functions, i.e.,... 

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