The self-consistent field procedure

From current estimates of P(1s;r) and P(2p;r), functions Y(nl;r) and X(nl;r) can be computed. The diagonal energy parameter can be estimated using the Rayleigh Quotient. The latter is obtained by multiplying eq:hf by P(nl;r), integrating, dividing by $\langle nl \vert nl \rangle$ (which is unity in our case) to yield

$$\varepsilon_{nl,nl} = \langle nl \vert {\cal L} \vert nl \ran......rac{2}{r}\left[Y(nl;r)\mbox{$P(nl;r)$ }+ X(nl;r)\right] \,dr .$$ (16)

Then the differential equation has the form

$$\left(\frac{d^2}{dr^2} + f(r) -\varepsilon_{nl,nl}\right)P(nl;r) = g(r)$$ (17)

with boundary conditions P(nl;0) =0 and $P(nl;r)\rightarrow0$ as $r \rightarrow\infty$. As long as g(r) is not identically zero, solutions satisfying the boundary conditions exist for all values of $\varepsilon_{nl,nl}$. However, it should be remembered that the Hartree-Fock equations for the $1s3p\;\Term 3 P/$ state, for example, are exactly the same as those for the lowest 1s2p 3 P/, that the radial equations are eigenvalue problems with many solutions. In solving the coupled systems of equations, it is then important to control convergence to the desired solution. The Hartree-Fock program achieves this by node counting.

Specifically, the number of changes of sign in the solution P(nl;r) should be n-l-1. In practice, node counting is somewhat of an art since, particularly initially when the potential and exchange may be quite ``non-physical'', spurious oscillations my be present and we will see later that even the exact solutions may have oscillations in the large ror ``tail'' region of a function. Not as much is known about f-orbitals: because of their r⁴ behaviour near the origin (see Eq. eq:origin), small oscillations have been observed in this region but they usually disappear as the iterations converge.
 
 

Table: Hartree-Fock calculation for $1s2p\;\Term 3 P/$

 
 
 

Table:hf.log file for $1s2p\;\Term 3 P/$

Ex1 shows a Hartree-Fock calculation for $1s2p\;\Term 3 P/$ using the interactive Hartree-Fock program HF. The program starts with a few prompts for describing the nature of the problem to be solved. ATOM is a convenient label for the case (up to six characters) and does not affect the computation. TERM designates the coupling and hence the energy expression defining the stationary conditions, and Z is the nuclear charge. In many cases there will be filled subshells and these can be specified separately from the 1s(1)2p(1) open subshells. It is possible to perform calculations where some orbitals are fixed, so the answer to the next prompt specifies the functions to be varied. The program sets a number of default parameters which can be overridden by user input, whenever needed. At this point, the program has all the information needed to attempt to perform the calculation.

Initial estimates are formed for the two radial functions as screened hydrogenic functions with effective nuclear charge

$$Z_{eff} = Z - \sigma(nl),$$ (18)

where $\sigma(nl)$ is a screening constant . The hf.log file shown in Ex1log informs us that the initial screening constants were zero for 1s and unity for 2p, in agreement with an intuitive concept of 1s being an inner orbital and 2p an outer orbital. The calculation then proceeds to compute the potential, exchange, and diagonal energy parameter ED for each radial function in turn. Tabulated is the value of ED, AZ$= \mbox{$P(nl;r)$ }/r^{l+1}$ as $r \rightarrow0$ eq:origin, the normalization integral of the computed solution, NORM, and a quantity related to the maximum change in the new solution over the previous estimate, DPM (Froese Fischer 1986). Since the radial functions are assumed to be normalized to unity, as the solutions converge, NORM should approach unity and clearly the DPM should approach zero. After solving the equations in turn, the program solves the least self-consistent orbital a few times before starting a new iteration.

The SCFTOL is a measure of the degree to which the system is expected to be self-consistent. Rather than failing some high expectation, the program relaxes this criterion so that lower self-consistency is met. If the desired degree of self-consistency is not met, the calculation can be ``recycled''. A file wfn.out was formed by HF: by moving (or renaming) this file to wfn.inp and running HF again, higher accuracy may be achieved.

After the iterations have converged, the program can be requested to compute additional information. We will not describe this option here. Further details are found in Appendix D.

The hf.log file provides more information about the nature of the solution. Some of the orbital parameters are printed: the diagonal energy parameter, E(nl)$=\varepsilon_{nl,nl}$; I(nl)$\equiv I(nl,nl)$; the kinetic energy, KE(nl), defined as

$$E^{\mbox{kinetic}}(nl) = -\frac{1}{2} \int_0^\infty\mbox{$P(n......{d^2}{dr^2} -\frac{l(l+1)}{r^2}\right]\mbox{$P(nl;r)$ }\, dr;$$ (19)

a relativistic shift correction (described later); the effective screening constant $\sigma(nl)$ for the final solution denoted by S(nl) in the computer output; and AZ(nl).

Hartree-Fock radial functions differ from screened hydrogenic functions. Even so, it is useful to define an effective screening constant for a Hartree-Fock function in that it provides some global information about the function. Screening constants can be defined in many ways, but Hartree (1957) suggested that the screening constant be such that

$$\langle r\rangle _{\mbox{\tiny HF}} = \langle r\rangle_{Z_{{eff}}}$$ (20)

where $\langle r\rangle_{Z_{eff}}$ is the mean radius of a hydrogenic atom with Z= Z_eff. In other words, the screening parameter is defined such that the mean radius of the Hartree-Fock orbital and that of a screened hydrogenic orbital is the same. Ex1log shows that the screening of the 1sis indeed small but that the 1s does not totally screen the nucleus for the 2p.

The next set of parameters are the expectation values of the moments as indicated, with ${\ \tt Delta}({\tt R})$ the expectation of $\frac{1}{4\pi}\delta(r)r^{-2}$. Following this is total energy information. The total energy, the relativistic shift, and the resulting relativistic energy is printed. In addition, the non-relativistic energy is divided into the kinetic energy and the potential energy. For an exact calculation, the Virial Theorem (Cowan 1981)  requires that the ratio of the potential energy to kinetic energy be exactly -2, and therefore ${ \tt Total\ Energy} =-{ \tt Kinetic\ Energy}.$ Deviations from these relationships are indications of numerical inaccuracy, lack of convergence, or that some of the orbitals were fixed.

2001-01-09