Matrix element integration

Effect of Diatom Stretching Dependence. The features of the poten-tial energy surface most central to a discussion of its effect on the predissociation process are not the individual radial strength functions V j((R), but rather the vibrational matrix elements (integrated over the diatom bond length) of the full potential... 

Consider the addition of matrix-element integrals. Suppose C = A+ B. A typical matrix element of C in the /j basis is... 

The Fock Matrix Elements. To solve the Roothaan equations (13.157), we first must express the Fock matrix elements (integrals) in terms of the basis functions x-The Fock operator F is given by (13.149), and... 

Under the assumption that the matrix elements can be treated as constants, they can be factored out of the integral. This is a good approximation for most crystals. By comparison with equation Al.3.84. it is possible to define a fiinction similar to the density of states. In this case, since both valence and conduction band states are included, the fiinction is called the joint density of states ... 

The //yj matrices are, in practice, evaluated in temis of one- and two-electron integrals over the MOs using the Slater-Condon mles [M] or their equivalent. Prior to fomiing the Ffjj matrix elements, the one-and two-electron integrals. 

Next, we shall consider four kinds of integrals. The first is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at that nucleus. The second is the expectation value of the Coulomb potential by one nucleus for one of the primitive basis function centered at a different point (usually another nucleus). Then, we will consider the matrix element of a Coulomb term between two primitive basis functions at different centers. The third case is when one basis function is centered at the nucleus considered. The fourth case is when both basis functions are not centered at that nucleus. By that we mean, for two Gaussian basis functions defined in Eqs. (73) and (74), we are calculating... 

In fact, the Coulomb integrals discussed in Section IV.C are available in contemporary quantum chemistry packages. We do not really need to develop our own method to calculate them. However, it is necessary to master the algebra so that we can calculate the matrix elements of the derivatives of the Coulomb potential. In the following, we shall demonstrate the evaluation of these matrix elements. 

Obviously, this matrix element is zero due to the integral over z. Similarly, we know that... 

The ti eatment of the Jahn-Teller effect for more complicated cases is similar. The general conclusion is that the appearance of a linear term in the off-diagonal matrix elements H+- and H-+ leads always to an instability at the most symmetric configuration due to the fact that integrals of the type do not vanish there when the product < / > / has the same species as a nontotally symmetiic vibration (see Appendix E). If T is the species of the degenerate electronic wave functions, the species of will be that of T, ... 

If Lhc live iiul c pun lien i oiic-ceiiicr iwo-cIccLroii integrals are expressed by symbols such as Gss, Gsp, defiiietJ above, then the Fock matrix element contributions from the one-center two-elec-iron in icgrals are ... 

In a closed-shell system, P = P) = P and the Fock matrix elements can be obtained by making this substitution. If a basis set containing s, p orbitals is used, then many of the one-centre integrals nominally included in INDO are equal to zero, as are the core elements Specifically, only the following one-centre, two-electron integrals are non-zero (/x/x /x/x), (pit w) and (fti/lfM/). The elements of the Fock matrix that are affected can then be written a." Uxllow s ... 

All sueh matrix elements, for any one- and/or two-eleetron operator ean be expressed in terms of one- or two-eleetron integrals over the spin-orbitals that appear in the determinants. 

The second term in the above expansion of the transition dipole matrix element Za 3 if i/3Ra (Ra - Ra,e) can become important to analyze when the first term ifi(Re) vanishes (e.g., for reasons of symmetry). This dipole derivative term, when substituted into the integral over vibrational coordinates gives... 

In this form, it is elear that E is a quadratie funetion of the Cl amplitudes Cj it is a quartie funetional of the spin-orbitals beeause the Slater-Condon rules express eaeh <

Cl matrix element in terms of one- and two-eleetron integrals < > and... 

As presented, the Roothaan SCF proeess is earried out in a fully ab initio manner in that all one- and two-eleetron integrals are eomputed in terms of the speeified basis set no experimental data or other input is employed. As deseribed in Appendix F, it is possible to introduee approximations to the eoulomb and exehange integrals entering into the Foek matrix elements that permit many of the requisite Fj, y elements to be evaluated in terms of experimental data or in terms of a small set of fundamental orbital-level eoulomb interaetion integrals that ean be eomputed in an ab initio manner. This approaeh forms the basis of so-ealled semi-empirieal methods. Appendix F provides the reader with a brief introduetion to sueh approaehes to the eleetronie strueture problem and deals in some detail with the well known Hiiekel and CNDO- level approximations. 

However, E is a quartic function of the Cy,i coefficients because each matrix element <

involves one- and two-electron integrals over the mos ( )i, and the two-electron integrals depend quartically on the Cyj coefficients. The stationary conditions with respect to these Cy i parameters must be solved iteratively because of this quartic dependence. 

---