Many-electron system

The total wavefimction for an electron in the hydrogen atom, or in any other electronic system, must result from a combination of atomic orbitals and spin wavefunctions. The appropriate combination is a product because the probabilistic interpretation requires that a total wavefimction must be zero whenever any of its parts is zero. The product function is called a spin-orbital. For example, the single electron in the Is orbital of hydrogen may be described by two such spin-orbitals  

Not considering the rather improbable hydrogen anion, helium is the smallest system with more than one electron around the nucleus. The helium wavefimction must depend on the coordinates of both electrons, formally labeled 1 and 2. The kinetic energy operators are as usual, and there are now three coulombic terms, representing the interaction of each of the two electrons with the nucleus, now with charge +2, and the electrostatic interaction between the two electrons, at a distance rn- The SchrSdinger equation is 

This differential equation cannot be solved analytically. The insurmountable difficulty is that the e /ri2 term depends on the coordinates of both electrons, and there is no way of breaking down the total wavefunction into a radial and an angular part. 

A very productive approximation, one that has been successfully used even for complex molecules, is to build complex wavefunctions from combinations of simple spin-orbitals. Consider again the Is orbital for the hydrogen atom, now with two electrons in it, and let (li)i and (ls)2 be the orbital written in the coordinates of electrons 1 and 2, respectively. There are four possible spin-orbitals  

The total wavefunction to be constructed from these basis functions might then be made of sums or products of spin-orbitals. But there is a new prescription, coming again from mathematical speculation based on well established experimental facts, having to do with the multiplicity of atomic spectral lines. This prescription is known 

As long as one is interested only in the total energy of the atomic electron system, the change from the simple but unrealistic PNC to a roughly realistic FNC is much more important than finer details due to variation of the finite nucleus model. This can be seen also from a recently published comparative study on numerical Dirac-Hartree-Fock calculations for 

A remark should be made here with respect to the generation and adjustment of the widely used effective core potentials (ECP, or pseudopotentials) [85] in standard non-relativistic quantum chemical calculations for atoms and molecules. The ECP, which is an effective one-electron operator, allows one to avoid the explicit treatment of the atomic cores (valence-only calculations) and, more important in the present context, to include easily the major scalar relativistic effects in a formally non-relativistic approach. In general, the parameters entering the expression for the ECP are adjusted to data obtained from numerical atomic reference calculations. For heavy and superheavy elements, these reference calculations should be performed not with the PNC, but with a finite nucleus model instead [86]. The reader is referred to e.g. [87-89], where the two-parameter Fermi-type model was used in the adjustment of energy-conserving pseudopotentials. 

The study of the electronic structure of diatomic species, which can nowadays be done most accurately with two-dimensional numerical finite difference techniques, both in the non-relativistic [90,91] and the relativistic framework [92-94], is still almost completely restricted to point-like representations of the atomic nuclei. An extension to allow the use of finite nucleus models (Gauss-type and Fermi-type model) in Hartree-Fock calculations has been made only very recently [95]. This extension faces the problem that different coordinate systems must be combined, the spherical one used to describe the charge density distribution p r) and the electrostatic potential V(r) of each of the two nuclei, and the prolate ellipsoidal one used to solve the electronic structure problem. 

While one-electron systems provide a good formal testing ground for an approximate theory, for quantum chemistry we need a theory that encompasses many-electron systems. Formally, we can treat the regular approximations as a Foldy-Wouthuysen transformation with a particular choice of X, and then we can write the transformed two-electron operator as 

The operator X is given in (18.34) which we repeat here without the subscript  

This operator is very similar to the 2 operator of the free-particle Foldy-Wouthuysen transformation it has a regularizing factor multiplied by ( r p). We can derive spin-free and spin-dependent operators from the Coulomb, Gaunt, and Breit interactions in an entirely analogous fashion to the Foldy-Wouthuysen transformation. As an example, the two-electron spin-orbit interaction in the regular approximation is 

This operator is complicated by the inclusion of the regularization terms, which must be differentiated in addition to l/r,-y. 

The biggest problem that we now face is that the potential V is in the denominator, and we have to decide what to include in V. Obviously, including gij = 1 /r,y would make the two-electron terms impossibly complicated, even if we were tabulating V on a grid. We are therefore forced to make some kind of approximation. 

No-pair energies (eV) of the 2s — 2p3/2 transition in Li-like uranium. Bq and are first-order frequency-independent and frequency-dependent Breit energies, respectively. Bo X Bn and x B are corresponding higher-order Breit energies, respectively. 

In Table 10, RCI [76] and MBPT [85] energies on the 2s—2pi/2 and Is — lpoji transitions in Li-like ions are compared with experiment. For these low- to imd-Z ions, higher-order Breit corrections are quite negligible and RCI and MBPT are in very good agreement 

Theoretical and experimental energies (eV) for the 2s — 2pi/2 and 2s — 2ps/2 transitions in Li-like ions. References to these results can be found in [76]. 

The strength of the RCI method is that it is intrinsically an all-order method as long as the Cl expansion is saturated with enough configurations. This is further demonstrated in Fig. 16 which shows another comparison with MBPT on the energies of the 2s — 

2s2p Pi transition in Be-like ions. In the nonrelativistic Z-expansion theory, the transition energy between levels of the same principal quantum numbers (An = 0) are given by a 1/Z expansion series 

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For many-electron systems such as atoms and molecules, it is obviously important that approximate wavefiinctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore, they should be antisynnnetric with respect to interchange of each pair of electrons. Such states can always be constmcted as linear combinations of products such as... 

QMC teclmiques provide highly accurate calculations of many-electron systems. In variational QMC (VMC) [112, 113 and 114], the total energy of the many-electron system is calculated as the expectation value of the Hamiltonian. Parameters in a trial wavefiinction are optimized so as to find the lowest-energy state (modem... 

Perdew J P and Zunger A 1981 Self-interaction correction to density-functional approximations for many-electron systems Phys. Rev. B 23 5048... 

NtofUer C and M S Plesset 1934. Note on an Approximate Treatment for Many-Electron Systems. Physical Review 46 618-622. 

Perdew J P and A Zunger 1981. Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems. Physical Review B23 5048-5079. 

Electronic Connectors. The complexity and size of many electronic systems necessitate constmction from relatively small building blocks which ate then assembled with connectors. An electronic connector is a separable electrical connector used in telecommunications apparatus, computers, and in signal transmission and current transmission <5 A. Separable connectors ate favored over permanent or hard-wired connections because the former facilitate the manufacture of electronic systems also, connectors permit assemblies to be easily demounted and reconnected when inspection, replacement, or addition of new parts is called for. 

Petersson and coworkers have extended this two-electron formulation of asymptotic convergence to many-electron atoms. They note that the second-order MoUer-Plesset correlation energy for a many-electron system may be written as a sum of pair energies, each describing the energetic effect of the electron correlation between that pair of electrons ... 

Bohr s treatment gave spectacularly good agreement with the observed fact that a hydrogen atom is stable, and also with the values of the spectral lines. This theory gave a single quantum number, n. Bohr s treatment failed miserably when it came to predictions of the intensities of the observed spectral lines, and more to the point, the stability (or otherwise) of a many-electron system such as He. 

In a recent paper Ostrovsky has criticized my claiming that electrons cannot strictly have quantum numbers assigned to them in a many-electron system (Ostrovsky, 2001). His point is that the Hartree-Fock procedure assigns all the quantum numbers to all the electrons because of the permutation procedure. However this procedure still fails to overcome the basic fact that quantum numbers for individual electrons such as t in a many-electron system fail to commute with the Hamiltonian of the system. As aresult the assignment is approximate. In reality only the atom as a whole can be said to have associated quantum numbers, whereas individual electrons cannot. 

Our earlier discussion of off-diagonal s, however, indicated that 2Q would substantially exceed in many-electron systems. Consequently, if Eq. 34 is used with N an empirical factor,31 we may expect the N values to exceed substantially the actual number of electrons of the outer subshell. 

This completes the discussion of the theory for many-electron systems from which we believe we have elucidated the reason for the failure of earlier approximate formulas for London forces. Many approximations remain and we summarize a few that seem most. important. 

Table II shows clearly the large differences between various theories for many-electron systems. The Kirkwood-Muller equation always yields somewhat too large coefficients for the atoms which are the only spherical systems but the London equation deviates by a greater amount on the low side. The Slater-Kirkwood equation gives a high value for He but yields coefficients smaller than the empirical ones for all other cases.

The main difficulty in solving the Schrodinger equation (Eq. II. 1) for a many-electron system comes from the two-electron interaction terms... 

For two-electron systems (He, H2) the method with different orbitals for different electrons was thoroughly discussed at the Shelter Island Conference in 1951 (Kotani 1951, Taylor and Parr 1952, Mulliken 1952). A generalization of this method to many-electron systems has now been given (Lowdin 1954, 1955, Itoh and Yoshizumi 1955) and is called the method with different orbitals for different spins. 

The main advantage of the method with correlation factor, based on Eq. III. 128 or Eq. III. 129, lies in the fact that it may be applied to any many-electron system. The practical calculation of the energy integrals involved may be fairly cumbersome, but the approach is nevertheless straightforward. 

A weakness of the development in the literature up to now has been that too much effort has been concentrated on the helium problem, whereas more complicated systems have been only scarce-ly treated. The reason is obvious it is much easier to test a new method for treating correlation on the ground state of helium, and if the method fails on this simple system, it will certainly not work on a more complicated system either. In treating energy differences in many-electron systems, simple methods will often produce results in excellent agreement with experiment owing to a fortuitous cancellation of errors, but a test on helium will then often reveal the faults of the approach. Even in the future, one can therefore expect that the helium problem will be paid a great deal of interest. 

The main method so far for treating correlation in many-electron systems is based on an expansion of configurations of the type III.18 ... 

The plasma model itself gives an important contribution to the theory of systems containing highly mobile electrons, and particularly its treatment of the screening phenomena is of value. The model has been carefully described in some reviews, and here we would like to refer to Pines (1955). We note that the plasma model has essentially been constructed for treating metals, but it would be interesting to see whether the basic ideas could be applied also to other many-electron systems. 

The purpose of this bibliography is to give a brief survey of the development of the methods for treating the correlation effects in many-electron systems by listing the most important papers in this field year by year. In accordance with the general outline used in Part I, Section III.C, the following methods will be included ... 

M0LLER, Chr., and Plesset, M. S., Phys. Rev. 46, 618, Note on an approximation treatment for many-electron systems."... 

Magnetic heat capacity of nickel, 133 Magnetic susceptibility, 25 Maleic anhydride, 168 Many electron system, correlations in, 304, 305, 318, 319, 323 Melting temperature and critical temperature for disordering correlation, 129... 

APPENDIX A-MOLECULAR ORBITAL TREATMENT OF MANY-ELECTRON SYSTEMS... 

Hamiltonian operator, 2,4 for many-electron systems, 27 for many valence electron molecules, 8 semi-empirical parametrization of, 18-22 for Sn2 reactions, 61-62 for solution reactions, 57, 83-86 for transition states, 92 Hammond, and linear free energy relationships, 95... 

A sum-over-states expression for the coefficient A for the expansion of the diagonal components faaaa was derived by Bishop and De Kee [20] and calculations were reported for the atoms H and He. However, the usual approach to calculate dispersion coefficients for many-electron systems by means of ab initio response methods is still to extract these coefficients from a polynomial fit to pointwise calculated frequency-dependent hyperpolarizabiiities. Despite the inefficiency and the numerical difficulties of such an approach [16,21], no ab initio implementation has yet been reported for analytic dispersion coefficients for frequency-dependent second hyperpolarizabiiities which is applicable to many-electron systems. 

For a many electron system, eq.(8) can be cast into the following dyadic form B "(R, Ro, B) =... 

Kryachko E. S., Ludena E. V., "Energy Density Functional Theory of Many-Electron Systems" Kluwer, Dordrecht, 1990. 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

Dirac, P. A. M. 1929 Quantum mechanics of many-electron systems. Proc. R. Soc. Land A 123, 714-733. 

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