Single-particle function

The total wavefunction r2,. . ., r is written as a product of single-particle functions (Hartree approximation). The various integrals are evaluated in tire saddle point approximation. A simple Gaussian fomr for tire trial one-particle wavefunction... 

The occupied single-particle functions (j) and the virtual single-particle functions ( )a are solutions of the corresponding canonical HF equations... 

Let us consider now the application of local-scaling transformations to sets of single-particle functions or orbitals. As it was shown in Sect. 2.1, a set of plane waves gives rise to the transformed orbitals described by Eq. (2). In particular, the application of this transformation to one-dimensional plane-waves leads to Harriman s equidensity orbitals [27], which are given by ... 

Let us now consider the transformed orbitals from the perspective of localscaling transformations. For a fixed set of single-particle functions <, (r) and a fixed set of expansion coefficients Cr, the wavefunction given by Eq. (114) is also fixed and yields a one-particle density p (r). We now consider the localscaling transformation that carries this density into a density p(r) and obtain the corresponding transformed orbitals ... 

Atomic units are used. Here and in the following x = (r, s) stands for the combined spatial and spin coordinates, r and s, respectively. The SOs 0,(x) constitute a complete orthonormal set of single-particle functions. 

Equations of motion for the time-dependent coefficients Aj time-dependent single particle functions, and time-dependent Gaussian parameters A K s) = aj c s), f r]jK s can be derived via the Dirac-Frenkel variational principle [1], leading to... 

Exercise. The basic reason why the fn are more useful than the Qn for describing dots is the following. Most quantities A in whose average one is interested are sum functions , i.e., they consist of a single-particle function a(xa) summed over all particles, or of a pair function fl(tff,v) summed over all pairs etc. In general,... 

V a(C) are the basis single-particle functions, we remind, that spin quantum numbers are included in a, and spin indices are included in = r, a as variables. 

The wavefunction F° then follows as an antisymmetrized product built from the single-particle functions q>i(r, ms) for the Z electrons (Slater determinantal wave-function, see below and Section 7.2), where r is the spatial vector and ms the spin magnetic quantum number. 

A special case of equ. (7.41a) is the /-coupling of single-particle functions ... 

In order to calculate the matrix elements with the Coulomb operator Vc, one again uses Slater determinantal wavefunctions, for the intermediate state xp(Mp, t) as well as for the complete final state which contains the doubly charged ion, f, and the two ejected electrons, x<, (Ka, Kb). Assuming that there is no correlation between the two escaping electrons and that their common boundary condition applies separately to each single-particle function, the directional emission property is included in the factors f( ka) and f( kb), and one gets for this Coulomb matrix element C... 

If this functional is varied with respect to the single particle functions one obtains the Dirac-Fock-Slater equations... 

The MCTDH method [79-82,113,114] uses a time development of the wavefunc-tion expanded in a basis of sets of variationally optimized time-dependent functions called single-particle functions (SPFs). A set of SPFs is used for each particle, where each particle represents a coordinate or a set of coordinates called combined mode. Indeed, when some modes are strongly coupled, and when there are many degrees of freedom, it is more efficient to combine sets of coordinates together as a... 

The recent discovery of ceramic high-Tc superconductors has forced a re-examination of the basic concepts and physical assumptions employed in current theoretical approaches. In reexamining basic concepts, it is well to remember that the true N-electron wave function may be expanded in terms of components each of which is made up of N single particle functions and that this expansion can be made in (at least) two different ways ... 

Non-relativistic quantum theory of atoms and molecules is built upon wave-functions constructed from antisymmetrized products of single particle wave-functions. The same scheme has been adopted for relativistic theories, the main difference now being that the single particle functions are 4-component spinors (bispinors). The finite matrix method approximates such 4-spinors by writing... 

The actual numerical values for V (0) are obtained easily from the entries in Table 1, and provide lower boundaries for eigenvalues of single-particle functions, whenever one of these nuclear models is used. 

This missing symmetry provided a great puzzle to theorists in the early part days of quantum mechanics. Taken together, ionization potentials of the first four elements in the periodic table indicate that wavefunctions which assign two electrons to the same single-particle functions such as... 

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