Superposition of configurations

From the density-matrix elements in Table 5.1, we find that the energy of the two-configuration state (5.2.13) is given by the expression 

To arrive at the best approximation to the electronic ground state in the minimal basis, we invoke the variation principle and minimize (5.2.28) with respect to r. Differentiating the energy with respect to r and setting the result equal to zero, we obtain  

The two-configuration energy (5.2.28) is periodic in r with period n. We choose tq and T in (5.2.31) as representing the ground and excited states since 

Our observations so far may be summarized by stating that, whereas the bonding and antibonding character of the one-electron density determines the coarse structure of the electronic system such as the arrangement of the electrons relative to the nuclei (accumulation and depletion of electrons in the internuclear region), the covalent and ionic character of two-electron density determines the finer details such as the arrangement of the electrons relative to each other (electron correlation). 

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Lemma 2 follows immediately from the additivity of rule R90, since any configuration is a superposition of configurations emerging from the initial states with a single nonzero site. The proof of Lemma 3 is left as an exercise for the reader. Note, however, that the first part of Lemma 3 shows that when N = 2 all configurations must eventually evolve to a fixed point consisting of all zero sites, since... 

Mainly for considerations of space, it has seemed desirable to limit the framework of the present review to the standard methods for treating correlation effects, namely the method of superposition of configurations, the method with correlated wave functions containing rij and the method using different orbitals for different spins. Historically these methods were developed together as different branches of the same tree, and, as useful tools for actual applications, they can all be traced back to the pioneering work of Hylleraas carried out in 1928-30 in connection with his study of the ground state of the helium atom. 

The three basic methods introduced by Hylleraas in his work on the He series have in modern terminology obtained the following names (a) Superposition of configurations (b) Correlated wave functions (c) Different orbitals for different spins. The first two approaches are developed almost to the full extent, whereas the last method is at least sketched in the 1929 paper. 

The method of superposition of configurations described in the first paper (1928) implies that, after choosing a complete basic set of one-electron functions rpk rx) one can develop the space function in Eq. III. 1 in the form ... 

Because of the success of the r12 method in the applications, one had almost universally in the literature adopted the idea of the necessity of introducing the interelectronic distances r j explicitly in the total wave function (see, e.g., Coulson 1938). It was there-fore essential for the development that Slater,39 Boys, and some other authors at about 1950 started emphasizing the fact that a wave function of any desired accuracy could be obtained by superposition of configurations, i.e., by summing a series of Slater determinants (Eq. 11.38) built up from a complete basic one-electron set. Numerical applications on atoms and molecules were started by means of the new modern electronic computers, and the results have been very encouraging. It is true that a wave function delivered by the machine may be the sum of a very large number of determinants, but the result may afterwards be mathematically simplified and physically interpreted by means of natural orbitals.22,17... 

On the helium problem, the connection between the method of correlated wave function and the method of superposition of configurations has also been investigated in detail.8... 

The method of superposition of configurations as well as the method of different orbitals for different spins belong within the framework of the one-electron scheme, but, as soon as one introduces the interelectronic distance rijt a two-electron element has been accepted in the theory. In treating the covalent chemical bond and other properties related to electron pairs, it may actually seem more natural to consider two-electron functions as the fundamental building stones of the total wave function, and such a two-electron scheme has also been successfully developed (Hurley, Lennard-Jones, and Pople 1953, Schmid 1953). 

In the three following sections we will try to sketch the mathematical foundation for the three approaches which are most closely connected with the Hartree-Fock scheme, namely the methods of superposition of configurations (a), correlated wave functions (b), and different orbitals for different spins (c). We will also discuss their main physical implications. 

Arbitrary Complete Basic Set The mathematical background of the method of superposition of configurations is quite simple and straightforward, and we will... 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

To test the accuracy and convenience of the method of superposition of configurations, the problem of the ground state of the helium atom has recently been reexamined by several authors. According to Hylleraas (1928), the total wave function may be expressed in the form... 

TABLE VI. Ground State of the Helium Atom Obtained by Superposition of Configurations ... 

The results in Table VI are of considerable interest also for atoms with more than two electrons, since they show the possibilities and limitations of the method of superposition of configurations/ when the latter are built up from one-electron functions... 

For atoms with more than two electrons, it is very difficult to obtain such a small absolute error in the energy as in the helium case, but, within an isoelectronic sequence, the relative error will, of course, go down rapidly with increasing atomic number Z. The method of superposition of configurations has been used successfully in a number of applications, particularly by Boys (1950-) and Jucys (1947-), and, for a more detailed survey of the work on atoms, we will refer to the special table on atomic calculations in the bibliography. This is a field of rapid development, where one can expect important new results within the next few years. 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

We note that it is possible to combine the method with correlation factor with the method using superposition of configurations to obtain any accuracy desired by means of comparatively simple wave functions. For a very general class of functions g(r12), one can develop the quotient (r r2)lg(r12) according to Eq. III.2 into products of one-electron functions y>k(r), which leads to the expansion... 

Hylleraas, E. A., Z. Physik 48, 469, "Uber den Grundzustand des Heliumatoms." Superpositions of configurations. 

Shull, H., and Lowdin, P.-O., J. Chew. Phys. 23, 1362, "Role of the continuum in superposition of configurations." Criticism of Taylor and Parr (1952). Emphasis of the work by Hylleraas (1928). 

Berry, R. S., J. Chew. Phys. 26, 1660, -electron structure of butadiene. Several methods including superposition of configurations. 

Fain, J., and Matsen, F. A., J. Chem. Phys. 26, 376, Complete -electron treatment of the butadiene molecule and ion." Complete VB. Results in agreement with SCF with superposition of configurations. 

Lowdin, P.-O., and Redei, L., Combined use of the methods of superposition of configurations and correlation factor on the ground states of He-like ions."... 

Water, electronic correlation, 324 methane-propane ternary system, 23 superposition of configuration, 295 Wigner s theory, cellular method," 252, 304, 306, 318... 

In the classical picture developed above, the wavepacket is modeled by pseudoparticles moving along uncorrelated Newtonian trajectories, taking the electrons with them in the form of the potential along the trajectory. In this spirit, a classical wavepacket can be defined as an incoherent (i.e., noninteracting) superposition of configurations, %i(R, t) /,-(r, f)... 

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