The Hartree-Fock solutions for $1s2s\;\Term 3 S/$ and 1 S/ states in He

The $1s2s\;\Term 3 S/$ and 1 S/    states in Helium are similar in that both have the same configuration and only the coupling of the spin momenta differs. This has an unexpectedly large effect on the Hartree-Fock approximation and the ease with which the HF equations can be solved. Whereas the 3 S/ state is the lowest of its symmetry the 1 S/ has the same symmetry as $1s^2\;\Term 1 S/$ and so the Hartree-Fock solution for $1s2s\;\Term 1 S/$ is a stationary state and not an energy minimum. It is an approximation to the second 1 S/ eigenvalue of the Hamilton. The two approximations differ in the way the energy varies with respect to a rotation of the orbital basis.

The energy expression for this system is

$${\cal E}(1s2s) = I(1s,1s)+I(2s,2s) +F^{0}(1s,2s) \pm G^{0}(1s,2s)$$ (30)

where the + sign applies to the 1 S/ state, and the - to the 3 S/. The space part of the configuration states is proportional to (see eq:CSFper)

$$\P{1s}{1}\P{2s}{2}\pm \P{2s}{1}\P{1s}{2}.$$ (31)

Consider a rotation of the radial basis to new basis. Let ${\bss P}$ be the column vector [P(1s;r), P(2s;r)]^t and let ${\bss P}^* = {\bss O}{\bss P}$, where ${\bss O}$ is an orthogonal matrix. Then ${\bss P} = {\bss O}^t {\bss P}^*$ or

$$\left[\begin{array}{c} P(1s;r)\\ P(2s;r) \end{array} \rig... ...begin{array}{c} P^*(1s;r)\\ P^*(2s;r) \end{array} \right].$$

For 3 S/ it is easy to show that the functional form of the energy expression does not change, that the energy itself is the same whether computed in the original or transformed basis. But it is even simpler to show that the wave function does not change. Writing the the space part in terms of the rotated basis, after a little algebra (or use of a symbol manipulation package) and using the fact that a² +b² =1, it follows that the form for 3 S/ remains the same.

A similar study for 1 S/ shows that the form of the energy expression changes. The transformed space part, in fact, is then a linear combination of the space parts of configuration state, $\Phi(1s2s\;\Term 1 S/)$ and the function $\{\Phi(1s^2\;\Term 1 S/) - \Phi(2s^2\;\Term 1 S/)\}$. In the limit of high Z, as will be shown in the next chapter, these two basis states are degenerate and this is the cause of the numerical problems associated with the computation of the Hartree-Fock solution for $1s2s\;\Term 1 S/$.

In the $1s2s\;\Term 3 S/$ case, the Hartree-Fock radial functions are not unique unless another condition is introduced. Koopmans (1933) suggested that the desirable transformation was the one that also minimized the core. More precisely, he showed that a transformation of the radial functions transformed the Lagrange multipliers, that the physically desirable solution was the one for which the diagonal energy parameters assumed extreme values, and that this solution was characterized by a zero value for the off-diagonal Lagrange multiplier. Thus the definition of the Hartree-Fock solution requires off-diagonal Lagrange multipliers to be zero whenever the wave function is invariant with respect to rotations of the radial basis.

  

Table: Hartree-Fock calculations for $1s2s\;\Term 1 S/$ for Helium

The Hartree-Fock solution for $1s2s\;\Term 1 S/$ can be obtained provided some critical steps are followed. First a calculation should be performed for the rapidly convergent $1s2s\;\Term 3 S/$ state and the wave function file wfn.out moved to wfn.inp to be used as initial estimates for the singlet state. Ex3 shows part of the calculation. Notice that now one of the default values needs to be changed: the STRONG parameter should be set to true so that after every orbital update, results are considered for orthogonalization. This is called strong orthogonality compared to the weak orthogonality assumed in earlier examples. Of course, the orbital that is orthogonalized should be less self-consistent than the orbital to which it is orthogonalized. With these two special steps, the calculation converges, although the virial theorem shows that results are not as accurate as found in previous cases.

As soon as input data describing the problem has been obtained, the program rotates the orbital basis for a stationary energy (to first order). The orbitals are updated. If the current orbital is less self-consistent than the other, it is orthogonalized, At the end of an iteration, all orbitals are Schmidt orthogonalized.

  

Figure: Comparison of the $1s2s\;\Term 3 S/$ and 1 S/ Hartree-Fock radial functions

The rotation has a significant effect on the radial functions which are shown in fig1s2s. Those for 3 S/ have the expected hydrogenic form but the rotation greatly reduces the height of the first maximum in the P(2s;r) for $\Term 1 S/$ and introduces a node into the P(1s;r) function for this state.