The spatial functions

The Pauli exclusion principle requires that the total wave function for electrons (fermions) have the property 

Thus the singlet spatial function is symmetric and the triplet one antisymmetric. If we use the variation theorem to obtain an approximate solution to the ESE requiring symmetry as a subsidiary condition, we are dealing with the singlet state for two electrons. Alternatively, antisymmetry, as a subsidiary condition, yields the triplet state. 

We must now see how to obtain useful solutions to the ESE that satisfy these conditions. 

---

The best way to ensure controlled development of the brief, and of the cost plans which necessarily accompany it, is to set up a system of data sheets which define, in increasing detail as the project develops, the spatial, functional and servicing requirements of each component part of the project. A well-resourced technical client may wish to do this himself. More commonly, it is a task which should be entrusted to the design team. Whoever does it, regular joint review of this information will help to ensure satisfaction with the finished product. 

Thus, the spatial function (q) is actually a set of eigenfunctions t/ n(q) of the Hamiltonian operator H with eigenvalues E . The time-independent Schrodin-ger equation takes the form... 

Suppose there are In electrons in the system, half with a-spin and half with -spin, and we form a single-determinant wavefunction with the spatial functions and so that... 

When the Coulomb term is entirely neglected, substitution into (1) shows that (t)> will be a solution provided the spatial functions satisfy... 

We are now done with spin functions. They have done their job to select the correct irreducible representation to use for the spatial part of the wave function. Since we no longer need spin, it is safe to suppress the subscript in Eq. (5.110) and all of the succeeding work. We also note that the partition of the spatial function X is conjugate to the spin partition, i.e., 2"/ , 2. From now on, if we have occasion to refer to this partition in general by symbol, we will drop the tilde and represent it with a bare X. 

We have so far said little about the nature ofthe space function, S. Earlier we implied that it might be an orbital product, but this was not really necessary in our general work analyzing the effects of the antisymmetrizer and the spin eigenfunction. We shall now be specific and assume that S is a product of orbitals. There are many ways that a product of orbitals could be arranged, and, indeed, there are many of these for which the application of the would produce zero. The partition corresponding to the spin eigenfunction had at most two rows, and we have seen that the appropriate ones for the spatial functions have at most two columns. Let us illustrate these considerations with a system of five electrons in a doublet state, and assume that we have five different (linearly independent) orbitals, which we label a, b,c,d, and e. We can draw two tableaux, one with the particle labels and one with the orbital labels. 

Letting Ax denote the standard deviation of the spatial function and Aw the standard deviation of the spectral function, we must necessarily have... 

Many of the most effective constraints set well-defined limits to the data function (or its spectrum) beyond which the correct function is not allowed to go. An important example of this type of constraint is nonnegativity, whereby the correctly restored function is not allowed to extend below the zero baseline and thereby take on nonphysical negative values. This is an appropriate constraint for spectroscopy and optical images. A further example of the constraints of fixed limits is that of an upper bound to the values of the restoration. Another important constraint of this type is that of finite extent, for which no deviations from zero are allowed for the spatial function over those intervals on the spatial axis that lie outside the known... 

We discovered in Chapter 9 that the spatial function as given by the discrete Fourier transform (DFT) is a discrete Fourier series. Letting u(k) denote the (known) series consisting of only low-frequency terms and v(k) the series consisting of only high-frequency terms, we want to determine the unknown coefficients in v(k) that best satisfy the constraints. Expressing deviations of the total function forbidden by the constraints as some function of u(k) + v(k), we shall try to determine the coefficients of v(k) that minimize these deviations. Sum-of-squares expressions for these measures of the error have been found to result in the most efficient computational schemes. 

The number of additional Fourier spectral components to recover is the option of the researcher. The number of iterations to execute with the most general computer program written is also the option of the researcher. A tolerance is presently used to determine the number of iterations performed. However, it is found in practice that only 5 or 10 iterations yield sufficiently accurate results for nearly all experimental data of interest. With the presently used computer program, restoration is to the spatial function, and the improved spatial function and the improved values of the coefficients are both generated with each iteration. If the improved Fourier spectrum is not desired, then additional computational time could be saved by neither reading nor writing the Fourier coefficients. When M data points are treated, the computer memory requirements are seldom more than 1M words. If it is not necessary to determine the extended Fourier spectrum, then more than 5M words are seldom needed in computer memory. 

Note that the second and third integrals on the r.h.s. are zero because of the orthonormality of the spatial orbitals a and b, whose products appear over the same electronic coordinate in those integrals. The spatial functions integrate to one in the first and fourth integrals, and the remaining spin expectation values are just diose of Eq. (C.17). Thus, the expectation value of Eq. (C.23) is (1 — 0 — 0+1) = 1. With additional work, it can be shown that 50 50q, not an eigenfunction of S-. 

The spatial functions (1.249) and (1.250) must be multiplied by a spin function. For electrons, we have the following four possible two-electron spin functions, three symmetric and one antisymmetric ... 

Unfortunately, the spatial functions of the activity coefficient and of the transference numbers in the general junction are not known and as a result (6.22) cannot be integrated. This situation becomes much worse when the junction connects two different solvents. In such cases the resulting measurement error from undefined Ej can reach several hundred millivolts. 

For the integrations in ab initio calculations we need the actual mathematical form of the spatial functions, and the hydrogenlike expressions are Slater functions [1]. For atomic and some molecular calculations Slater functions have been used [3]. These vary with distance from where they are centered as exp(-constant.r), where r is the radius vector of the location of the electron, but for molecular calculations certain integrals with Slater functions are very time-consuming to evaluate, and so Gaussian functions, which vary as exp(-constant.r2) are almost always used a basis set is almost always a set of (usually linear combinations of) Gaussian functions [4]. Very importantly, we are under no theoretical restraints about their precise form (other than that in the exponent the electron coordinate occurs as exp(-constant.r2)). Neither are we limited to how many basis functions we can place on an atom for example, conventionally carbon has one 1 s atomic orbital, one 2s, and three 2p. But we can place on a carbon atom an inner and outer Is basis function, an inner and outer 2s etc., and we can also add d functions, and even f (and g ) functions. This freedom allows us to devise basis sets solely with a view to getting... 

In our previous treatment of helium, we only used the spatial function 1 s (1) 1 s (2) to calculate the energy, while ignoring the spin part. This is acceptable because energy is independent of spin. 

Now we can consider the symmetry properties of the spatial functions corresponding to the above spin functions. They are uniquely defined from the requirement that the product of the spatial and spin functions must be antisymmetric. In a way, what was symmetrized in the spin part (rows) must be antisymmetrized in the spatial part (columns) and vice versa. That means that the spin function represented by a two-row Young pattern T with the first row longer by 2S boxes than the second one must be complemented by a spatial function represented by a two-column Young pattern T with the first column longer by 2S boxes than the second one, e.g. ... 

Consequently, for the total wave function to be properly antisymmetric, the spatial function to be multiplied by the spin functions must be symmetric or antisymmetric for singlet or triplet states, respectively. Satisfying these requirements may be made more explicit in the following way. 

The spatial functions sk are orthonormal and are each individually eigenfunctions of H with eigenvalue E ... 

The dimension of the irreducible representation D A S() is fA. Note that since H contains no spin interaction terms, the operations dl affect only the spatial functions in equation (31) and not the spin functions. 

The approximation (33) corresponds to taking as the spatial function 0 in equation (18), the product form... 

In the macroscopic theory of electromagnetic waves [3], the evanescent wave (EW) arises from the requirement that the boundary conditions be satisfied at all points on the flat (ideal) interface between two materials of different optical properties that are uniform throughout the materials. The spatial functions in the exponents describing propagation of plane waves in each material are set equal... 

Fig 4 The spatial function d(r) for a single single molecule form-factor fit and after removal of the cluster contribution for a-ice. 

Usually, the. spatial function ilt is constructed from the summation of one-electron spatial orbitals (atomic orbitals) 4>. known as the basis set. u.sed to construct a MO. This approach is known as the LCAO method (/inear combination of utomic orbitals). It is an approximation of the accurate many-elcetron wave function (Eq. 28-54). The atomic orbital contributions are weighted by coefficients c,. The summation is truncated, so the ip function is not complete, which has consequences when. solving for E. 

When periodic boundary conditions are used, the spatial functions /)/(r) are given by Eq. (3.58). Because we look for real solutions of the Maxwell equation A takes the fonn analogous to the corresponding real classical solution... 

We now consider the expansion of the spatial function O in a set of orthonormal one-electron functions. For this purpose we can employ not a single orthonormal set but a different set for each electron coordinate ... 

---