Matrix Hamiltonian

We now show what happens if we set up tire Hamiltonian matrix using basis functions i ), tiiat are eigenfiinctions of Fand with eigenvalues given by ( equation A1.4.5) and (equation Al.4.6). We denote this particular choice of basis fiinctions as ij/" y. From (equation Al.4.3). (equation A1.4.5) and the fact that F is a Hemiitian operator, we derive... 

That is, in the basis rj.F.irjjthe Hamiltonian matrix is block diagonal in Fand and we can rewrite (equation A1.4.8) as... 

The Hamiltonian matrix factorizes into blocks for basis functions having connnon values of F and rrip. This reduces the numerical work involved in diagonalizing the matrix. 

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. 

The Hamiltonian matrix will be block diagonal in this basis set. There will be one block for each irreducible representation of the synnnetry group. 

Hamiltonian matrix H in this basis set is a matrix with elements given by the integrals... 

The eigenvalues E of It can be detemiined from the Hamiltonian matrix by solving the secular equation... 

The vanishing integral rule is not only usefi.il in detemiining the nonvanishing elements of the Hamiltonian matrix H. Another important application is the derivation o selection rules for transitions between molecular states. For example, the hrtensity of an electric dipole transition from a state with wavefimction "f o a... 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The conclusion is therefore that the 4x4 Hamiltonian matrix, which is assumed to have zero trace, takes the fonn... 

The two surface calculations by using the following Hamiltonian matrix ai e rather stiaightfoiTvard in the diabatic representation... 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

Since the form of the electronic wave functions depends also on the coordinate p (in the usual, parametric way), the matrix elements (21) are functions of it too. Thus it looks at first sight as if a lot of cumbersome computations of derivatives of the electronic wave functions have to be carried out. In this case, however, nature was merciful the matrix elements in (21) enter the Hamiltonian matrix weighted with the rotational constant A, which tends to infinity when the molecule reaches linear geometry. This means that only the form of the wave functions, that is, of the matrix elements in (21), in the p 0 limit are really needed. In the above mentioned one-elecbon approximation... 

The approach developed by Jungen and Merer (JM) [24] is of a similar level of sophistication. The main difference is that IM prefer to remove the coupling between the electronic states by a transformation of the Hamiltonian matrix (i.e., vibronic energy matrix), rather that of the Hamiltonian itself. They first calculate the large amplitude bending functions for one of the adiabatic potentials, as if it belonged to a E electronic state. These functions are used as... 

I h e preceding discussion mean s that tli e Matrix etjuatiori s already described are correct, except that the Fuck matrix, F. replaces the effective one-electron Hamiltonian matrix, and th at K depends on th e solution C ... 

In an ab initio method, all the integrals over atom ic orbital basis function s are com puted and the bock in atrix of th e SCK com puta-tiori is formed (equation (6 1) on page 225) from the in tegrals. Th e Kock matrix divides inui two parts the one-electron Hamiltonian matrix, H, and the two-electron matrix, G, with the matrix ele-m en ts... 

In the Kx ten ded H tick el approx irn ation, lli c eh urges in th c u n selected part arc treated like classical point charges. The correction of these classical charges to the diagonal elcincntsof the Hamiltonian matrix may be written as ... 

In these eases, one says that a linear variational ealeulation is being performed. The set of funetions Oj are usually eonstrueted to obey all of the boundary eonditions that the exaet state E obeys, to be funetions of the the same eoordinates as E, and to be of the same spatial and spin symmetry as E. Beyond these eonditions, the Oj are nothing more than members of a set of funetions that are eonvenient to deal with (e.g., eonvenient to evaluate Hamiltonian matrix elements I>i H j>) and that ean, in prineiple, be made eomplete if more and more sueh funetions are ineluded. 

Symmetry provides additional quantum numbers or labels to use in describing the mos. Each such quantum number further sub-divides the collection of all mos into sets that have vanishing Hamiltonian matrix elements among members belonging to different sets. 

By parameterizing the off-diagonal Hamiltonian matrix elements in the following overlap-dependent manner ... 

Sinee there are only two SALC-AOs and both are of different symmetry types these SALC-AOs are MOs and the 2x2 Hamiltonian matrix reduees to 2 1x1 matriees. 

So, there are IA2", 2A and 2E orbitals. Solving the Hamiltonian matrix for eaeh symmetry bloek yields ... 

The full N terms that arise in the N-eleetron Slater determinants do not have to be treated explieitly, nor do the N (N + l)/2 Hamiltonian matrix elements among the N terms of one Slater determinant and the N terms of the same or another Slater determinant. 

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