Molecular eigenvalue equation

The equations may be simplified by choosing a unitary transformation (Chapter 13) which makes the matrix of Lagrange multipliers diagonal, i.e. Ay 0 and A This special set of molecular orbitals (f> ) are called canonical MOs, and they transform eq. (3.40) mto a set of pseudo-eigenvalue equations. 

This looks like an eigenvalue equation, but is not, since instead of regenerating fa, a sum of functions 4>j is obtained. For the complete set of molecular orbitals the equation can be written in matrix notation as... 

This is an eigenvalue equation for the Floquet exponents and the coefficients To see this more clearly, let us introduce the basis of states an), where the Greek letter labels the molecular states and the Roman letter the Fourier components. Then, Eq. (8.19) can be written as... 

One of the aspects of the molecular orbital approach which is most interesting is that the eigenfunctions of the operator h are apparently not too different from the final atomic eigenfunctions. We shall discuss this point further, but for the moment we simply note that the eigenfunctions 0, (r,R) satisfy the eigenvalue equation... 

A useful feature of the molecular orbital approach is that the eigenvalue equation of Eq. (23.22) can be separated in confocal elliptic coordinates,23 and, equally important, these eigenfunctions are apparently somewhat similar to the final atomic eigenfunctions.22 The coordinates are given by22... 

We want an eigenvalue equation because (cf. Section 4.3.4) we hope to be able to use the matrix form of a series of such equations to invoke matrix diagonalization to get eigenvalues and eigenvectors. Equation (5.35) is not quite an eigenvalue equation, because it is not of the form operation on function = k x function, but rather operation on function = sum of (k x functions). However, by transforming the molecular orbitals to a new set the equation can be put in eigenvalue form (with a caveat, as we shall see). Equation 5.35 represents a system of equations... 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

Moreover, using the eigenvalue equation (A.l) of the molecular Hamiltonian, and then suppressing the closeness relation between the dipole moment operator, the SD is transformed into... 

The first two terms are the molecular Hamiltonian and the radiation field Hamiltonian. The molecular Schrodinger equation for the first term in (5.2) is assumed solved, with known eigenvalues and eigenfunctions. Solutions for the second term in (3.4) in vacuo are taken in second-quantized form. Hint can be taken in minimal-coupling form (5.3) allowing for the variation of the radiation field over the extent of the molecule,... 

Matrix-eigenvalue equation. 103-105 Molecular diagrams. 119-121 Molecular orbitals as a linear combination of atomic orbitals, 8-9... 

Solve the Fock matrix eigenvalue equations given above to obtain the orbital energies and an improved occupied molecular orbital. In so doing, note that the normalization condition <0i i> = 1 = ]SCi gives the needed normalization condition for the expansion coefTicients of the 0i in the atomic orbital basis. 

For a molecule of N atoms with its structure at a local energy minimum, the normal modes can be calculated from a 3A x 3A1 mass-weighted second derivative matrix H, the Hessian matrix, defined in a molecular force field such as CHARMM [32-34] or AMBER [35-38]. For each mode, the eigenvalue X and the 3A x 1 eigenvector r satisfy the eigenvalue equation. Hr = Ar. 

The molecular orbital coefficients (/, /i = 1,2, , m) which specify the nature, and hence, energy of the orbital ///, arc determined by solving the eigenvalue equation of the effective one-electron Hamiltonian,associated with the molecule (equation 1.11) ... 

Accordingly, the molecular orbitals defined by the eigenvalue equation H = e

molecular orbitals also as adiabatic states. The atomic orbitals are obtained from HQ = where Hq stands for the Hamiltonians Ha = T+ and... 

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