Excited states even-tempered basis sets

EVEN-TEMPERED BASIS SETS FOR EXCITED STATES... 

In this section, we turn to the approximation of the excited state in a finite basis set. In particular, we describe three different schemes for developing systematic sequences of even-tempered basis sets for excited states which will approach the exact solution in the limit M 00. We define each scheme, which we label (a), (6) and (c) in turn. 

The ground and excited state matrix Hartree-Fock energies for the He, Li and Be atoms are presented in Tables 1, 2 and 3, respectively. All energies are given in atomic units, (Hartree). In each of these tables, we label the columns according to the three schemes, (a), (b) and (c), described above for generating sequences of even-tempered basis sets. We consider each system in turn. 

TABLE 1. Self-Consistent Field energies (in hartree) of the He atom for some excited states as a function of the size, M, of the even-tempered basis set used to parametrize the orbitals. In the column headed (a) the same even-tempered basis set optimized for the ground state, is used for aU states ... 

It has been demonstrated that for the excited states of the atoms He, Li and Be considered in the present work, a simple optimization of the a and [3 parameters for each size of basis set leads to a sequence of even-tempered basis sets capable of supporting high accuracy in Hartree-Fock calculations for excited state energies of atoms. Furthermore, optimization of the a and f3 parameters for the smallest basis set in a sequence, M = 6 in the present study, followed by application of the recursion formulae (40) and (41) represents a good compromise which undoubtedly proved useful in case where full optimization of these parameters for each size of basis set is computationally demanding. 

The parameters a and were optimized for each atom and a given excited state. Information of the even-tempered basis sets for low-lying states of the He and Li atom can be found in Ref. [47]. More information for highly excited state basis sets is available from authors on request. 

EXCITED STATE SELF-CONSISTENT FIELD THEORY USING EVEN-TEMPERED PRIMITIVE GAUSSIAN BASIS SETS... 

In the scheme which we label (a), the same basis set is employed for both the ground and excited state. Therefore, the same integrals over basis functions are used for both states. The values of the even-tempered parameters a and /3 for M = 6 are those which were optimized for the ground state of the atom as reported by Schmidt and Ruedenberg [9]. These values are given in Table 1 of reference [9j. Larger basis sets were generated by means of the recursion formulae [10] ... 

In scheme (6) the basis set is optimized by invoking the variation principle for each state considered. For the ground state the optimized values of the even-tempered parameters ao and / o given by Schmidt and Ruedenberg [9] are used. We add the subscript 0 to distinguish ground state values. For the excited state optimal ai and / i values for a sequence of Mi values are determined. 

TABLE 4. Optimized even-tempered parameters a and / for excited states of He as a function of size of basis set. 

In Tables 8 and 9, the even-tempered parameters for the two excited states of the beryllium atom considered in this work are presented, with Table 8 giving the parameters obtained according to scheme (b) in which the parameters were fully optimized and Table 9 giving the parameters obtained by means of scheme (c) m which the parameters a and (3 were only optimized for the smallest basis set, i.e. M = 6. 

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