Wavefunctions, momentum density

The momentum wavefunction is associated with the momentum density yi(p) through the relation... 

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

We restrict ourselves to the clamped-nucleus or Born-Oppenheimer approximation [30,31] because essentially all the work done to date on electron momentum densities has relied on it. Therefore we focus on purely electronic wavefunctions and the electron densities that they lead to. 

All real-valued wavefunctions and the overwhelming majority of other approximate wavefunctions also produce [156] momentum densities that satisfy... 

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

It is apparent that only a trickle of work has been, and is currently being, done on momentum densities in comparison with the torrent of effort devoted to the position space electron density. Moreover, much of the early work on II( p) has suffered from an undue emphasis on linear molecules. Nevertheless, some useful insights into the electronic structure of molecules have been achieved by taking the electron momentum density viewpoint. The most recent phenomenal developments in computer hardware, quantum chemical methods and software for generating wavefunctions, and visualization software suggest that the time is ripe to mount a sustained effort to understand momentum densities from a chemical perspective. Readers of this chapter are urged to take part in this endeavor. 

G(R,Rq) is not known explicitly (or by quadrature) for any but the most simple (and uninteresting problems). But it is clearly related to the solution of a diffusion problem for a particle starting at Rq in a 3N dimension space and subject to absorption probability V(R) + Vq per unit time. We therefore expect to be able to sample points R from G(R,Rq) conditional on Rq. It turns out to be possible by means of a recursive random walk in which each step is drawn from a known Green s function for a simple subdomain of the full space for the wavefunction. References ( 4) and ( ) contain a thorough discussion of this essential technical point, and also of the methods which permit the accurate computation of the energy and other quantum expectations such as the structure function, momentum density, Bose-Einstein condensate fraction, and... 

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

Term wavefunctions describe the behaviour of several electrons in a free ion coupled together by the electrostatic Coulomb interactions. The angular parts of term wavefunctions are determined by the theory of angular momentum as are the angular parts of one-electron wavefunctions. In particular, the angular distributions of the electron densities of many-electron wavefunctions are intimately related to those for orbitals with the same orbital angular momentum quantum number that is. 

We have recently introduced the Wigner intracule (2), a two-electron phase-space distribution. The Wigner intracule, W ( , v), is related to the probability of finding two electrons separated by a distance u and moving with relative momentum v. This reduced function provides a means to interpret the complexity of the wavefunction without removing all of the explicit multi-body information contained therein, as is the case in the one-electron density. 

The r-space and p-space representations of the ( th-order density matrices, whether spin-traced or not, are related [127] by a fif -dimensional Fourier transform because the parent wavefunctions are related by a 3A -dimensional Fourier transform. Substitution of Eq. (5.1) in Eq. (5.8), and integration over the momentum variables, leads to the following explicit spin-traced relationship ... 

In addition to initial conditions, solutions to the Schrodinger equation must obey certain other constraints in form. They must be continuous functions of all of their spatial coordinates and must be single valued these properties allow VP P to be interpreted as a probability density (i.e., the probability of finding a particle at some position can not be multivalued nor can it be jerky or discontinuous). The derivative of the wavefunction must also be continuous except at points where the potential function undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This condition relates to the fact that the momentum must be continuous except at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

Figure 3.8. Molecular orbitals for the oxygen atom, with indication of their quantum numbers (main, orbital angular momentum and projection along axis of quantisation). Shown is the oxygen nucleus and the electron density (where it has fallen to 0.0004 it is identical for each pair of two spin projections), but with two different shades used for positive and negative parts of the wavefunction. The calculation uses density functional theory (B3LYP) and a Gaussian basis of 9 functions formed out of 19 primitive Gaussian functions (see text for further discussion). The first four orbitals (on the left) are filled in the ground state, while the remaining ones are imoccupied.

As described in Ref. [25], the Hartree approach has been applied to get energies and density probability distributions of Br2(X) 4He clusters. The lowest energies were obtained for the value A = 0 of the projection of the orbital angular momentum onto the molecular axis, and the symmetric /V-boson wavefunction, i.e. the Eg state in which all the He atoms occupy the same orbital (in contrast to the case of fermions). It stressed that both energetics and helium distributions on small clusters (N < 18) showed very good agreement with those obtained in exact DMC computations [24],... 

We calculate the roton [Steinhauer 2004] in atomic vapor BEC using a Jas-trow wavefunction, which allows us to calculate beyond mean-field correlations in the quantum state, giving rise to a roton peak in the structure factor at a momentum k oc a-1 (where a is the s-wave scattering length of the atoms). The relevant parameter e = J i t a (n is the density of the atoms in the BEC), is calculated over a range of values from e < 1 up to e 1. 

Spin-coupled wavefunctions have also recently been used in a critical appraisal of local density approximations in small molecules and for a comparison of the effects of bond formation on electron densities in momentum space and in position space. An example of a momentum space-orbital is shown in Fig. 12 for at 3.75 bohr in the range of R where the spin-coupling coefficients are undergoing large changes. The obvious diffraction effects are greatest in this region a detailed interpretation of momentum-space orbitals is presented in Ref. 81. 

Momentum-space concepts are not, in general, familiar to the chemist and so we outline first the calculation of momentum-space electron densities, p p), from ab initio wavefunctions. The form of pip) for different molecules is discussed, using as examples (i) the ground state of H2, (ii) bond formation in BH", and (iii) the n-orbitals in large conjugated polyenes. 

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