Zeroth-order eigenfunctions

Note that h is simply the diagonal matrix of zeroth-order eigenvalues In the following, it will be assumed that the zeroth-order eigenfunction a reasonably good approximation to the exact ground-state wavefiinction (meaning that Xfi , and h and v will be written in the compact representations... 

If the unperturbed system is degenerate, so that several linearly independent eigenfunctions correspond to the same energy value, then a more complicated procedure must be followed. There can always be found a set of eigenfunctions (the zeroth order eigenfunctions) such that for each the perturbation energy is given by equation 9 and the perturbation theory provides the... 

The system to be treated consists of two nuclei A and B and two electrons 1 and 2. In the unperturbed state two hydrogen atoms are assumed, so that the zeroth-order energy is 2WH-If the first electron is attached to nucleus A and the second to nucleus B, the zeroth-order eigenfunction is ip (1)

zeroth-order energy, so that the system... 

In case that the symmetry character of an electron-pair structure and an ionic structure for a molecule are the same, it may be difficult to decide between the two, for the structure may he anywhere between these extremes. The zeroth-order eigenfunction for the two bond electrons for a molecule MX (HF, say, or NaCl) with a single electron-pair bond would be... 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

If the small time interval in question extends from f to f + f then the matrix elements of the propagator for this short time interval may be written as a matrix in the basis of the zeroth-order eigenfunctions ... 

An alternate treatment, due to Dirac, uses the variation of the coefficients to discuss a system of many levels, not just two levels. In analogy to the above, use the perturbation expansion, Eq. (3.34.2), and the zeroth-order eigenfunctions vF (r, t) as follows ... 

Besides these zeroth-order eigenfunctions, we shall also be interested in the eigenfunctions of the total Hamiltonian X In what follows, we shall assume that the effects of the perturbation A.X are small enough that we can correlate the exact gri eigenstates with their parent zero-order eigenstate q thus we can useftdly retain the quantum numbers q to define the exact eigenstates. (This is convenient but not essential). Thus after including the effects of the perturbation A.X, we have... 

The zeroth-order eigenfunctions are identical to g " j and their energies are equal to Eg. At the first-order level the matrix ( kg / g -I-... 

The final chapter discusses the case when H2 dissociates into and H , i.e. the next higher-lying dissociation limit of H2. The undisturbed eigenfunction for the two-electron system (H ) is now ij/n or, equivalently, (p 2- HL discuss qualitatively the energy curves of the eigenvalues E and which belong to the zeroth-order eigenfunctions... 

Accordingly, the complete set of zeroth-order eigenfunctions is assumed to be known. The coefficient at the zeroth-order ground state in Eq. (12.4) is often chosen as Co = 1 which can be achieved by an appropriate renormalization of F. By this choice, Eq. (12.4) becomes ... 

That is, the Fermi vacuum is the zeroth order eigenfunction in the ground state. The corresponding eigenvalue is the sum of the energies of the occupied orbitals, and not the Hartree-Fock energy. 

With this additional consideration we can of course not derive the first order eigenfunctions from the secular problem, but we can derive the zeroth order eigenfunctions which are determined solely by the symmetry character of the differential equation. Instead of immediately substituting (17a) into the secular problem with H, it is more practical to first form certain linear combinations of (17a), namely (we leave out the normalisation factors)... 

If we view these two atoms (2) as unperturbed systems and let them interact with each other, then the problem of the correct zeroth order eigenfunctions of the total system and their symmetry characteristics arises. 

The problem of the exchange term in Hartree-Fock equations has been treated in different ways. The HF-Slater (HFS) method was used in [15]. Numerical SCF calculations of ground-state total energies in relativistic and nonrelativistic approximations are compared in [16, 17]. HFS wavefunctions served as zeroth-order eigenfunctions to compute the relativistic Hamiltonian. In [18], seven contributions to the total energy (including magnetic interaction, retardation, and vacuum polarization terms) are detailed. 

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