Wave equation determinantal

Slater developed an approach (the "determinantal method") that offers a way of choosing among linear combinations (essentially sums and differences) of the polar and nonpolar terms in the Hund-Mulliken equations to bring their method into better harmony with the nonpolar emphasis characteristic of the Heitler-London-Pauling approach in which polar terms do not figure in the wave equation. 72... 

The relation of this treatment to the perturbation theory of Chapter VI can be seen from the following arguments. If the functions F (x) were the true solutions (x) of the wave equation 27-1, the determinantal equation 27-7 would have the form... 

In the last chapter we have seen that a good approximation to the wave function for a system of atoms at a considerable distance from one another is obtained by using single-electron orbital functions wa(l), etc., belonging to the individual atoms, and combining them with the electron-spin functions a and /3 in the form of a determinant such as that of Equation 44-3. Such a function is antisymmetric in the electrons, as required by Pauli s principle, and would be an exact solution of the wave equation for the system if the interactions between the electrons and those between the electrons of one atom and the nuclei of the other atoms could be neglected. Such determinantal... 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

ABBA molecules, 631-633 HCCS radical, 633-640 perturbative handling, 641-646 theoretical principles, 625-633 Hamiltonian equation, 626-628 vibronic problem, 628-631 Thouless determinantal wave function, electron nuclear dynamics (END) ... 

Equations (33) can be solved numerically in an iterative manner, leading finally to self-consistency. In this way the energy Eqs and the determinantal wave function at minimum ESs are obtained and available for many systems from simple atoms to large molecules, neutral or ionized. 

It should be noted that in the case of the helium isoelectronic series (singlet GS of a two-electron system), because only one orbital is sufficient to construct the determinantal wave function and to obtain the density as n(r) — 2[t)r(r)], the minimization over n in Eq. (64) is equivalent to the minimization over ip in Eq. (28). Therefore the HF-KS and HF equations and their eigenfunctions are the same, ipir) = and consequently AT n =... 

Equation (A.10) (or (A.12)) has an inherent restriction built into it since other wave functions of the same form are possible if one could select any Ne orbitals from an infinite number of them rather than the Ne used in (A.10). One could thus generate an infinite number of determinantal wave functions of the form (A.10), and without approximation, the exact wave function E( 1,2,..., Ne) could be expressed as a linear combination of them ... 

Although in principle an exact solution to the Schrodinger equation can be expressed in the form of equation (A.13), the wave functions and coefficients da cannot to determined for an infinitely large set. In the Hartree-Fock approximation, it is assumed that the summation in equation (A.13) may be approximated by a single term, that is, that the correct wave function may be approximated by a single determinantal wave function , the first term of equation (A.13). The method of variations is used to determine the... 

In summary, to obtain a many-electron wave function of the single determinantal form [equation (A.12)] which will give the lowest electronic energy [equation (A.14) or (A.27)], one must use one-electron wave functions (orbitals) which are eigenfunctions of the one-electron Fock operator according to equation (A.42). There are many, possibly an infinite number of, solutions to equation (A.42). We need the lowest Ne of them, one for each electron, for equation (A. 12) [or (A.27)]. When the Ne MOs of lowest energy satisfy equation (A.42), then Eq=Ehf [equation (A.27)] and o= hf [equation (A.12)]. 

As it happens, if a molecule has the same number of electrons with spin up (a) as with spin down (ft), the solution of the HF equations in the vicinity of the equilibrium geometry and for the ground electronic state yields the result that the spatial part of the MOs describing a and ft electrons are equal in pairs. In other words, for the vast majority of molecules (F2 is an exception), the HF determinantal wave function may be written as... 

Notice that the energy of the HF determinantal wave function, equation (A.68), and for that matter for any single determinantal wave function, can be written by inspection Each spatial orbital contributes ha or 2h according to its occupancy, and each orbital contributes 2J — in its interaction with every other molecular orbital. Thus, the energy of the determinant for the molecular ion, M+, obtained by removing an electron from orbital of the RHF determinant, is given as... 

Earlier it was argued that the many-electron wave function (the true solution to the electronic Schrodinger equation) could be expanded in terms of an infinite series of single determinantal wave functions [Equation (A. 13)] ... 

Vc, an exchange-correlation term, Exc(p), and an external potential, [V , which arises primarily from nuclear-electron attraction but could include extramolecular perturbations, such as electric and magnetic fields. If the electronic wave function were expressed as a determinantal wave function, as in HF theory, then a set of equations functionally equivalent to the HF equations (A.40) emerges [324]. Thus... 

Density functional theory, 21, 31, 245-246 B3LYP functional, 246 Hartree-Fock-Slater exchange, 246 Kohn-Sham equations, 245 local density approximation, 246 nonlocal corrections, 246 Density matrix, 232 Determinantal wave function, 23 Dewar benzene, 290 from acetylene + cyclobutadiene, 290 interaction diagram, 297 rearrangement to benzene, 290, 296-297 DFT, see Density functional theory... 

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