Wave functions orthogonality

In the PP framework, the valence electron wave function % orthogonalized to the inner core electron wave function HA s is given by [12]... 

Note that the form of a CIS wave function differs from that of an ordinary Cl wave function. In an ordinary Cl wave function, the reference function (the SCF wave function for the state of interest) makes the largest contribution. In the CIS method for an excited state, the reference function is the SCF wave function for the ground state, and this reference function does not appear in the CIS wave function. (This makes the CIS wave function orthogonal to the ground-state wave function, which is desirable, so as to avoid having the variational calculation collapse to the ground state.) The CIS wave function includes only a modest amount of electron correlation. 

It is important to note that the two surface calculations will be carried out in the diabatic representation. One can get the initial diabatic wave function matrix for the two surface calculations using the above adiabatic initial wave function by the following orthogonal transformation,... 

U(qJ is referred to as an adiabatic-to-diabatic transformation (ADT) matrix. Its mathematical sbucture is discussed in detail in Section in.C. If the electronic wave functions in the adiabatic and diabatic representations are chosen to be real, as is normally the case, U(q ) is orthogonal and therefore has n n — l)/2 independent elements (or degrees of freedom). This transformation mabix U(qO can be chosen so as to yield a diabatic electronic basis set with desired properties, which can then be used to derive the diabatic nuclear motion Schrodinger equation. By using Eqs. (27) and (28) and the orthonormality of the diabatic and adiabatic electronic basis sets, we can relate the adiabatic and diabatic nuclear wave functions through the same n-dimensional unitary transformation matrix U(qx) according to... 

In modem quantum chemistry packages, one can obtain moleculai basis set at the optimized geometry, in which the wave functions of the molecular basis are expanded in terms of a set of orthogonal Gaussian basis set. Therefore, we need to derive efficient fomiulas for calculating the above-mentioned matrix elements, between Gaussian functions of the first and second derivatives of the Coulomb potential ternis, especially the second derivative term that is not available in quantum chemistry packages. Section TV is devoted to the evaluation of these matrix elements. 

A further simplification is made. The wave functions pi and p2, which are orthogonal and normalized in the hydrogen atom, are assumed to retain their orthonormality in the molecule. Orthonormality requires that... 

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

Traditionally, excited states have not been one of the strong points of DFT. This is due to the difficulty of ensuring orthogonality in the ground-state wave function when no wave functions are being used in the calculation. 

Computations done in imaginary time can yield an excited-state energy by a transformation of the energy decay curve. If an accurate description of the ground state is already available, an excited-state description can be obtained by forcing the wave function to be orthogonal to the ground-state wave function. 

State averaging gives a wave function that describes the first few electronic states equally well. This is done by computing several states at once with the same orbitals. It also keeps the wave functions strictly orthogonal. This is necessary to accurately compute the transition dipole moments. 

Finally, DFT methods are at present not well suited for excited states of the same symmetry as the ground state. The absence of a wave function makes it difficult to ensure orthogonality between the ground and excited states. 

SCVB wave functions to include electron correlation is due to the fact that the VB orbitals are strongly localized, and since they are occupied by only one electron, they have the built-in feature of electrons avoiding each other. In a sense, an SCVB wave function is tte best wave function that can be constructed in terms of products of spatial orbitals. By allowing the orbitals to become non-orthogonal, the large majority (80-90%) of what is called electron correlation in an MO approach can be included in a single determinant wave function composed of spatial orbitals, multiplied by proper spin cou ing functions. 

The primary feature of SCVB is the use of non-orthogonal orbitals, which allows a much more compact representation of the wave function. An MO-CI wave function of a certain quality may involve many thousand Slater determinants, while a similar quality VB wave function may be written as only a handful of resonating VB structures. 

A variation of wave function coefficients is subject to constraints like maintaining orthogonality of the MOs, and normalization of the MOs and the total wave function. 

If there is more than one constraint, one additional multiplier term is added for each constraint. The optimization is then performed on the Lagrange function by requiring that the gradient with respect to the x- and A-variable(s) is equal to zero. In many cases the multipliers A can be given a physical interpretation at the end. In the variational treatment of an HF wave function (Section 3.3), the MO orthogonality constraints turn out to be MO energies, and the multiplier associated with normalization of the total Cl wave function (Section 4.2) becomes the total energy. 

While it is not beyond the realm of possibility to evaluate the g s, h s, and Q for appropriate wave functions, some further simplification is necessary for our present arguments. We assume that the s are orthogonal to all y s. This is not necessarily true although the //s can be selected to make it true. Even without specific selection of /A s it is probably a good approximation for those s which contribute substantially to the polarizability. Then... 

In conclusion, we observe that many writers in the modern literature seem to agree about the convenience of the definition (Eq. 11.67), but that there has also been a great deal of confusion. For comparison we would like to refer to Slater, and Arai (1957). Almost the only exception seems to be Green et al. (1953, 1954), where the exact wave function is expanded as a superposition of orthogonal contributions with the HF determinant as its first term ... 

Let us now consider the possibilities for deriving an eigenfunction for a particular excited state. The straightforward application of the variation principle (Eq. II.7) is complicated by the additional requirement that the wave function Wk for the state k must be orthogonal to the exact eigenfunctions W0, Wv for all the lower states although these are not usually known. One must therefore try to proceed by way of the secular equation (Eq. III.21). A well-known theorem15 25 says that, if a truncated... 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

A possibly more accurate value for the double bond character of the bonds in benzene (0.46) id obtained by considering all five canonical structures with weights equal to the squares of their coefficients in the wave function. There is some uncertainty aS to the significance of thfa, however, because of- the noii -orthogOnality of the wave functions for the canonical structures, and foF chemical purposes it fa sufficiently accurate to follow the simple procedure adopted above. 

A possibly more reliable prediction can be made on the basis of Sherman s wave function for naphthalene,16 by considering all 42 canonical structures. The fractional double bond character of a bond can be considered to be given approximately (neglecting non-orthogonality of the canonical wave functions) by the expression... 

The wave functions for these resultant structures are mutually orthogonal. 

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