Zero-order wavefunction

A 6pn( state with an autoionization rate T is broadened, and its spectral density is described by a Lorentzian of width T centered at its energy Wq. A reasonable first approximation to the wavefunction is obtained by multiplying the zero order wavefunction of Eq. (19.6a) by the square root of a Lorentzian of width T ... 

Integration of this equation using the zero-order wavefunction yields an expression for the first derivative of the energy ... 

If the zero-order wavefunction does, in fact, satisfy Eqn. (26), then the first term on the right in Eqn. (32) is identically zero. This means that the first derivatives are obtained as an expectation value of the derivative Hamiltonian. This last statement is the Hellmann-Feynman theorem. 

As in perturbation theory, there is a 2n-l-l rule [45 7], Using the nth-order wavefunction, energy derivatives up to 2n+l may be evaluated directly, and the derivative wavefunctions between the n and 2n -F1 orders are not required explicitly. For example, with the first-derivative wavefunctions known explicitly, the third energy derivatives may be calculated directly. To see this for the abc derivative of Eqn. (38), one evaluates the equation at the equilibrium choice of parameters and then integrates with the zero-order wavefunction. That is,... 

In Standard Numerov-Cooley, integration or step-by-step use of Eqn. (101) begins both at close-in and at far-out extremes where the values of P and K are near zero, assuming a bound state. These are guessed to be very small values. Then, the inward and outward functions that are obtained are matched in slope and value at some midway point by iterative adjustment of the energy. In Eqn. (101), the zero-order energy E is known already and so the process requires no iteration. This also means the integration needs to be done in only one direction. The F that is found will be a mixture of the true derivative wavefunction and the zero-order wavefunction, and so the last step is a projection step to ensure orthonormality. 

We have found that the perturbation U couples each such eigenfunction to other zero-order wavefunctions according to (cf. Eq. (4.81))... 

In the absence of this coupling the two-molecule Hamiltonian is the sum of Hamiltonians of the individual chromophores. Consequently, the zero-order wavefunctions are products of terms associated with the individual molecules, for example,... 

States with Single Determinantal Zero-Order Wavefunctions... 

It is such types of heavy mixing that, together with additional evidence (see following sections), led to the proposal of opfimizing fhe quantitative description of electronic structures by computing appropriately chosen multiconfigurational Fermi-sea zero-order wavefunctions (Sections 3 and 8). 

COMMENTS ON FACTS FROM THE THEORY AND COMPUTATION OF GROUND STATES WITH SINGLE DETERMINANTAL ZERO-ORDER WAVEFUNCTIONS... 

These calculations and findings were made possible in the framework of the analyses that were published in the 1980s, for example. Refs. [60a, 60b, 61, 62]. Accordingly, the zero-order wavefunction was chosen as the direct, state-specific MCHF solution with only the intrashell configurations for each... 

This became possible not only by the state-specific nature of the computations but also by the realization that the natural orbitals produced from hydrogenic basis sets were the same as the MCHF orbitals that are computable for the intrashell states up to about N = 10 - 12. Therefore, for DES with very high N, instead of obtaining the multiconfigurational zero-order wavefunction from the solution of the SPSA MCHF equations (which are very hard to converge numerically if at all), we replaced the MCHF orbitals by natural orbitals obtained from the diagonalization of the appropriate density matrices with hydrogenic orbitals. 

The case of fhe ground state, S, was already discussed in Section 5, with numerical results for different values of Z. Bofh fhe H-F sea and the Fermi-sea zero-order wavefunctions are described by a superposition of fhe ls 2s IS and W2p SACs. 

Furthermore, and this is important for fhe inferprefafion of the bonding in Be2 to be discussed below, the results (17-20) show that primarily the state D and to a lesser degree the state carry with them a significant d — wave component in their Fermi-sea zero-order wavefunctions. This means that such orbitals should also be considered in the self-consistent, zero-order description of molecules containing Be. 

In symbols, the general compass is the form + 0 " of the wave-function of Eq. (8). In principle, the two parts are represented by different function spaces, whose elements and size depend on the problem. The zero-order wavefunction, is normally obtained self-consistently. Its orbitals belong to the state-specific Fermi-sea, see below. 

Due to the fact that at most single and double excitations can directly interact through the BO Hamiltonian with the zero-order wavefunction 0), it is sufficient to terminate the expansion at this level known as first-order interacting space ... 

In multireference perturbation theory, defining a proper zero-order Hamiltonian is anything but straightforward. The reference wavefunction, in general, is not an eigenfunction of the zero-order Hamiltonian. A second complication arises as interactions between the FOIS functions and zero-order wavefunction through the zero-order Hamiltonian cannot be excluded. Therefore, projection techniques are commonly employed. In NEVPT2, the zero-order Hamiltonian takes the form... 

The differences that are noted above imply that if an electron-correlation calculation were carried out using the same basis sets for the wavefunc-tions of these two spectroscopic terms, then the size requirements would be very large, simply in order to correct for the inadequacy of the zero-order wavefunctions. 

Suppose we consider a hydrogenic or a HF solution to an electronic structure that labels an autoionizing state. Let the symbol for this zero-order wavefunction be yo with energy eg. Either from Schrodinger or from Wigner-Brillouin perturbation theory, the expansion of the exact function shows that there are integrals of the form / over the continuum. So there... 

Fermi-vacua are determined separately. The linear combination of thus produced functions is constructed a posteriori, the weights taken from the unrelaxed zero-order wavefunction. 

Each zero-order wavefunction indicates that one electron occupies a Is and the other a 2s AO. Note that the first zero-order wavefunction <( 1,2) indicates that both electrons have a and the second that both have p spin. The third zero-order wavefunction indicates that the l5 electron has a spin and the 2s electron p spin while the fourth indicates that the Is electron has p while the 2s electron has a spin. 

Comparison with equation (2.39) shows that each of the two first integrals is equal E, the energy calculated for the zero-order wavefunction. Since the third integral is equal to... 

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