Diagonal determinant

A diagonal determinant is equal to the product of its diagonal elements. 

A determinant whose only nonzero elements occur in square blocks centered about the principal diagonal is in block-diagonal form. If we regard each square block as a determinant, then a block-diagonal determinant is equal to the product of the blocks. For example,... 

The diagonal of a determinant that runs from the top left to the lower right is the principal diagonal. A diagonal determinant is a determinant all of whose elements are zero except those on the principal diagonal. For a diagonal determinant. 

A diagonal determinant is a special case of a triangular determinant, so it also obeys this relation. 

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 eigenvalues of this mabix have the form of Eq. (68), but this time the matrix elements are given by Eqs. (84) and (85). The symmetry arguments used to determine which nuclear modes couple the states, Eq. (81), now play a cracial role in the model. Thus the linear expansion coefficients are only nonzero if the products of symmebies of the electronic states at Qq and the relevant nuclear mode contain the totally symmebic inep. As a result, on-diagonal matrix elements are only nonzero for totally symmebic nuclear coordinates and, if the elecbonic states have different symmeby, the off-diagonal elements will only... 

For states of different symmetry, to first order the terms AW and W[2 are independent. When they both go to zero, there is a conical intersection. To connect this to Section III.C, take Qq to be at the conical intersection. The gradient difference vector in Eq. f75) is then a linear combination of the symmetric modes, while the non-adiabatic coupling vector inEq. (76) is a linear combination of the appropriate nonsymmetric modes. States of the same symmetry may also foiiti a conical intersection. In this case it is, however, not possible to say a priori which modes are responsible for the coupling. All totally symmetric modes may couple on- or off-diagonal, and the magnitudes of the coupling determine the topology. 

Note that since the eigenfunctions are determined only up to an overall phase, the following hansformations also diagonalize 5R ... 

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

Thus in the lowest order approximation the angle x is eliminated from the off-diagonal matrix elements of [second and third of Eqs. (60)] it solely determines the selection rules for matrix elements of Hg with respect to nuclear basis functions. 

Next, we determine the conditions for this matrix to become diagonal (with numbers of norm 1 in the diagonal), which will happen if and only if when p and q fulfill the following relations ... 

If the nttmber of valence electrons thus calculated does not agree with the standard number of valence electrons in an atom, this atom carries a charge, in this case, the diagonal element h, has more or fewer valence electrons than the nominal value of the respective atom i. The charge value, Ah, can be determined by subtracting the sum of the row values from the nominal value (Eq, (3)). 

For any sizeable system the Slater determinant can be tedious to write out, let alone the equivalent full orbital expansion, and so it is common to use a shorthand notation. Various notation systems have been devised. In one system the terms along the diagonal of the matrix are written as a single-row determinant. For the 3x3 determinant we therefore have ... 

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