Basis wavefunctions

We consider systems containing IV 4 particles, ofrespective masses m,, with a spatial wavefunction depending on their positions Fi, . Fjv in a space-fixed coordinate system. These coordinates can be transformed into the position R of the system s center of mass and relative coordinates Fjjv,. ..,Fjv where r,-, stands for r — r we also define r,y = lr,yl and f,y = r,y/r,y. We restrict our attention to isolated systems in which it is not necessary to consider the motion of the center of mass, so the wavefunction can be expressed entirely in terms of the 31V — 3 relative coordinates, ofwhich three (two if IV = 2) can be chosen to specify the orientation of the entire wavefunction with respect to space-fixed axes, while the remaining coordinates can describe the internal stmcture of the wavefunction. The restriction to IV 4 causes the number of internal coordinates to be equal to the number of distinct r. Considering also that orbital angular momentum must be conserved, the two-particle system can have the trivial spatial wavefunction 

For a three-particle system, the orientation of the wavefunction depends upon both fi3 and f23, and P can be built from quantities such as 

Turning now to the four-body problem, a possible starting point would be the generalization of equation (2), 

Yan and Drake [19] proposed the use of these funetions in combinations yielding eigenfunctions of the total angular momentum of quantum numbers L, M, presenting their analyis in a form appropriate to the calculations by the Hylleraas method. 

At this point we observe that, even for the four-body case, only three coordinates are needed to define the spatial orientation of the wavefunction, and these have been determined once f 14 and f24 have been specified. It might, therefore, seem unnecessary to use a form as complicated as that in equation (5), and one might consider using an angular momentum eigenfunction such as 

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We first define basic elements of the model to, thereafter, construct basis wavefunctions finally, a quantized EM-field representation is used to further discuss the experimental results. 

Table V. Dipole Moments (/x) for LiH, Til and PbTe Calculated by Using Chemical Basis Wavefunctions ...

As an example, consider the interaction between the 3IIi state (cnr4Tr configuration) and the 3 + state (basis wavefunctions must have the same value of ft. For this example, the ft = 1 matrix elements of the L+S part of HSE will be evaluated. 

As already stated, the chemical isomer shifts of the ferricyanides differ little from those of the ferrocyanides, although they are generally smaller by about O l mm s for a given cation. Typical values are given in Table 7.2. They are, however, easily distinguished by the small quadrupole splittings displayed by the ferricyanides. The only example which has been extensively studied is K3[Fe" (CN)6], in which the quadrupole splitting is determined by the thermal population of three Kramers doublets. The latter are linear combinations of the I xf), I xz ) and yz y basis wavefunctions, and the appropriate... 

It is to be noted that the dependence of the electronic basis wavefunctions S/( rJ R ) on the nuclear coordinates R is a parametric one. Since the field-free electronic Hamiltonian is time independent, the basis set is independent of time. Yet, the expansion coefficients T /( R, f) in this basis are time-dependent and have to obey the reduced coupled Schrodinger equations ... 

The distributed multipole analysis (DMA) embodies a wavefimction partitioning method pioneered by Stone and coworkers [31, 108, 109], similar to a number of procedures developed by others [35, 101, 110]. DMA relies on the expansion of the charge density as a function of a product of basis wavefunctions... 

How does the first-order correction alterthe wavefunction Recall that the perturbation raises the potential energy near the top of the box (near L) much more than near the bottom (near x = 0) therefore, we expect the probability of finding the particle near the bottom to be enhanced compared with that of finding it near the top. Because the zero-order ground-state wavefunction is positive throughout the interior of the box, we thus expect the wavefunction itself to be raised near the bottom of the box and lowered near the top. In fact, the correcticHi terms do just this. First, note that the basis wavefunctions with odd n are symmetric with respect to the center of the box therefore, they would have the same effect near the top of the box as near the bottom. The coefficients of these terms are zero they do not contribute to the correction. The even- basis functions all start positive near x = 0 and end negative near x = L therefore, such terms must be multiplied by positive coefficients (as the result provides) to enhance the wavefunction near the bottom and diminish it near the top. 

Techniques for finding the best coefficients Cf for more complex molecules have been refined by comparing calculated molecular energies, dipole moments and other properties with experimental measurements, and a variety of software packages for such calculations are available [6, 23-30]. The basis wavefunctions employed in these descriptions usually are not the atomic orbitals obtained by solving the... 

Although the individual basis wavefunctions are not eigenfunctions of the full Hamiltonian, it is possible in principle to find linear combinations of these wavefunctions that do give such eigenfunctions, at least to the extent that the basis functions are a complete, orthonormal set. Equation (2.42) then becomes exact. The coefficients and eigenvalues are obtained by solving the simultaneous linear equations... 

If the basis wavefunctions and 2 are orthogonal and normalized, this expression reduces to... 

Equation (10.13) is a remarkably general expression. The density matrix p could refer to any system that can be described with a linear combination of basis wavefunctions. Further, A could represent the matrix of expectation values of the operator for any dynamic property. The only requirement, but an important one to note, is that the matrix elements of A and p must be expressed in terms of the same set of basis wavefunctions. 

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