Trial function

The representation of trial fiinctions as linear combinations of fixed basis fiinctions is perhaps the most connnon approach used in variational calculations optimization of the coefficients is often said to be an application of tire linear variational principle. Altliough some very accurate work on small atoms (notably helium and lithium) has been based on complicated trial functions with several nonlinear parameters, attempts to extend tliese calculations to larger atoms and molecules quickly runs into fonnidable difficulties (not the least of which is how to choose the fomi of the trial fiinction). Basis set expansions like that given by equation (A1.1.113) are much simpler to design, and the procedures required to obtain the coefficients that minimize are all easily carried out by computers. 

As formulated above in terms of spin-orbitals, the Hartree-Fock (HF) equations yield orbitals that do not guarantee that P possesses proper spin symmetry. To illustrate the point, consider the form of the equations for an open-shell system such as the Lithium atom Li. If Isa, IsP, and 2sa spin-orbitals are chosen to appear in the trial function P, then the Fock operator will contain the following terms ... 

This equation defines the Galerldn method and a solution that satisfies this equation (for aUj = 1,. . . , °°) is called a weak solution. For an approximate solution, the equation is written once for each member of the trial function, j = 1,. . . , NT — 1, and the boundary condition is apphed. 

The Galerldn finite element method results when the Galerldn method is combined with a finite element trial function. The domain is divided into elements separated by nodes, as in the finite difference method. The solution is approximated by a linear (or sometimes quadratic) function of position within the element. These approximations are substituted into Eq. (3-76) to provide the Galerldn finite element equations. The element integrals are defined as... 

FIG. 3-53 Trial functions for Galerldn finite element method linear polynomial on triangle. 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

In the relativistic KKR method the trial function inside the MT-sphere is chosen as a linear combination of solutions of the Dirac equation in the center-symmetrical field with variational coefficients C7 (k)... 

The definitions are here given under the assumption that the wave function XP is either antisymmetric or symmetric for a trial function without symmetry property, one has to replace the binomial factor NCV before the integrand by a factor l/p and sum over the N(N—l). . . (N—p+l) possible integrals which are obtained by placing the fixed coordinates x, x 2,. . ., x P in various ways in the N places of the first factor W and the fixed coordinates xv x2,. . xv similarly in the second factor W. By using Eq. II.8 we then obtain... 

Let us consider an arbitrary trial function f(xv x2,. . ., xN) without any symmetry properties at all. By means of the anti-symmetrization operator... 

The antisymmetrization of the trial function has a definite effect on the generalized density matrices P(x x 2.. . x x1x2. . . xv) defined by Eq. II.9 since, except for the first-order matrix, they will now all be antisymmetric in each set of the indices. For p = 2, we have in particular ... 

If a trial function 9 leads to a kinetic energy 1 and a potential energy Vx which do not fulfill the virial theorem (Eq. 11.15), the total energy (7 +Ei) is usually far from the correct result. Fortunately, there exists a very simple scaling procedure by means of which one can construct a new trial function which not only satisfies the virial theorem but also leads to a considerably better total energy. The scaling idea goes back to a classical paper by Hylleraas (1929), but the connection with the virial theorem was first pointed out by Fock.5 It is remarkable how many times this idea has been rediscovered and published in the modern literature. 

If = [Pg.220]

Let us then consider the case when cpx is an arbitrary trial function which does not satisfy the virial relation. From the relation Eq. 11.24 follows... 

For the sake of simplicity, we will consider a diatomic molecule with the internuclear distance R, but the result is directly general-izable to a system with several internuclear distances Rv R2,. In addition to the trial function = q>(rlt r2,. . ., rN, R), we will now also consider the scaled function ... 

Our derivation of Eq. 11.33 based on the use of the variation principle is different from Slater s original treatment, but so far follows Hirschfelder and Kincaid. Here we will now show that it also permits such a scaling of an arbitrary trial function internuclear distance according to Eq. 11.29. 

It seems as if an energy value of sufficiently high accuracy has now been found for the helium problem, but we still do not know the actual form of the corresponding exact eigenfunction. In this connection, the mean square deviation e = J — W 2 (dx) and criteria of the Eckart type (Eq. III.27) are not very informative, since s may turn out to be exceedingly small, even if trial function... 

From the point of view of principles, it is interesting to note that the method based on the generalized form of Eq. III. 129 seems to be very closely connected both with Wigner s classical theory described in Section III.B and with Bohm and Pines plasma model (Krisement 1957). Following Krisement, we will replace the various trial functions flt /2,. . ., fn in Eq. III.9 by a single average function /, and Wigner s basic wave function (Eq. II1.7) takes then the simple form... 

The variation condition 6At = 0 can be independently imposed for variations of and its adjoint. The condition of gauge invariance requires that trial functions have the form... 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

For Q = Q , this density function describes electronic motions for given nuclear positions, while for Q = Q it describes the quantal correlation of nuclear positions at time f, which should be small for classical-like variables. The equation of motion for the density function could be derived from the original LvN equation. Instead, it is more convenient to construct it from the wavefunctions. The phase factor and the preexponential factor are trial functions to be determined from the TDVP. The procedure followed here parallels that in ref. (23). 

The method presented here allows, starting with trial gaussian functions, a partial analytical treatment which we have used to improve the LCAO-GTO orbitals (trial functions) essentially obtained from all ab initio quantum chemistry programs. As in r-representation, trial functions (t>i( Hp) (Eq. 21) are conveniently expressed as linear combinations of m functions Xi(P) themselves written as linear combinations of Gt gaussian functions (LCAO-GTO approximation) gta(P). 

In all the variational methods, the choice of trial function is the basic problem. Here we are concerned with the choice of the trial function for the polarization orbitals in the calculation of polarizabilities or hyperpolarizabilities. Basis sets are usually energy optimized but recently we can find in literature a growing interest in the research of adequate polarization functions (27). 

Except for the restrictions stated above, the function 0, called the trial function, is completely arbitrary. If 0 is identical with the ground-state eigenfunction 00, then of course the quantity S equals Eq. If 0 is one of the excited-state eigenfunctions, then is equal to the corresponding excited-state energy and is obviously greater than Eq. However, no matter what trial function 0 is selected, the quantity W is never less than Eq. 

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