Wave function determinantal

The approximations defining minimal END, that is, direct nonadiabatic dynamics with classical nuclei and quantum electrons described by a single complex determinantal wave function constructed from nonoithogonal spin... 

Show that the atomic determinantal wave function... 

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. 

Density functional theory, direct molecular dynamics, complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 404-411 Density operator, direct molecular dynamics, adiabatic systems, 375-377 Derivative couplings conical intersections, 569-570 direct molecular dynamics, vibronic coupling, conical intersections, 386-389 Determinantal wave function, electron nuclear dynamics (END), molecular systems, final-state analysis, 342-349 Diabatic representation ... 

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) ... 

The determinantal wave function in Eq. (21) is built [23] from complex dynamical spin orbitals Even when the basis orbitals ut in Eq. (22) are orthogonal these dynamical orbitals are nonorthogonal, and for a basis of nonorthogonal atomic orbitals based on Gaussians as those in Eq. (24) the metric of the basis becomes involved in all formulas and the END theory as implemented in the ENDyne code works directly in the atomic basis without invoking transformations to system orbitals. 

Instead of a.o. s, as before, configurations (determinantal wave functions) are now interacting. Any number of configurations can be mixed in this way. 

The problem of T[p] is cleverly dealt with by mapping the interacting many-electron system on to a system of noninteracting electrons. For a determinantal wave function of a system of N noninteracting electrons, each electron occupying a normalized orbital >p, (r), the Hamiltonian is given by... 

The correlation problem alluded to in the Introduction has most often been approached through the use of determinantal wave functions, that is, by writing Peiec as a linear combination of Hartree-Fock-like functions, [6]... 

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 =... 

Here electron system with some local potential Us(r) [see Eqs. (36) and (40)]. Obviously, spin orbitals i x) - the solutions of Eq. (40) - and [Pg.74]

For density matrices generated from a determinantal wave function IF (which occurs, e.g., in the HF method or the KS approach), the p-th order matrix can be obtained from the 1-st order one... 

ORTHOGONALITY CONSTRAINTS FOR SINGLE DETERMINANTAL WAVE FUNCTIONS... 

We shall be concerned with ground and excited electronic states which can be adequately described by a single determinantal wave function, i.e. doublet states, triplet states, etc. with spin 5/0). Let 0 be the Slater determinant constructed from a set of spin-orbitals consisting of spatial part ( = 1,2,. ..,n") associated with a spin functions and orbitals... 

Another way that additional configurations can be added to the the ground-state wave function is by the use of Moller-Plesset perturbation theory (MPPT). As it happens, a Hamiltonian operator constructed from a sum of Fock operators has as its set of solutions the HF single determinantal wave function and all other determinantal wave... 

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... 

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)] ... 

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