2. Outline of TI-DFT and Its COEP Implementation
Over the years, a vast number of DFT papers devoted to excited states (ES) have been reported and only the scantiest of selections will be given here. We shall focus on TI-DFT-OEP based methods whose extension to excited states is neither unique nor straightforward. There exist the distinct formulations (see, e.g., [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29] and references therein) and their number continues growing. Some of them rely on a variational treatment focusing either on ensembles [
7,
18,
19,
20,
21,
22] or on an individual excited state approach [
3,
4,
13,
14,
15,
16,
17]. Others use non-variational approaches (e.g., [
25,
26,
27,
28,
29]).
An early rigorous DFT approach to treat excited states was given by Theophilou [
7] and was reformulated as a subspace DFT [
18] (later as equiensemble theory). It was later generalized into the theory of unequally weighted ensembles of excited states by Gross, Oliveria and Kohn [
19,
20,
21,
22]. An important step towards practical calculations within the ensemble theory was made by Nagy who generalized the OEP idea for ensemble of excited states [
23]. Later, a ghost-interaction correction to this scheme [
30] has showed on atoms that the ensemble Kohn–Sham theory with the exact ensemble-exchange potential can be as accurate as the ground state calculations [
30,
31]. However, practically no applications exist for highly excited states having the same spatial and spin symmetry.
For the universal variational excited-state functional of the kinetic and electron-electron repulsion energies, the second approach includes a “pure state” bifunctional formulation [
3,
4] or a “pure state” unifunctional formulation of Ayers, Levy, and Nagy [
15,
16] that utilizes special properties of the Coulomb potential. A useful analysis of the “pure state” approaches can be found in [
13,
14,
15,
16] and references within. An appeal of this pure-state approach is that it generates directly the desired individual excited-state energies and densities. This point was confirmed, to some extent, by excited state calculations based on the COEP method proposed in [
1] and later developed in Refs. [
32,
33,
34]. It was shown, that an excited state produced by excitation of electron, for example, from orbital
ϕ0k of the ground state KS determinant
Φ0 can be presented by imposition of some orthogonality constraints on the ES orbitals.
Let us consider a singly excited state to make clear the idea of this method. If the ground state orbitals are determined by a one-particle equation with an effective local potential
V0eff(r)
then orbitals {
ϕi} of the excited state determinant
Φ, generated by excitation of an electron from orbital
are determined by
subject to constraints
According to the asymptotic projection method constrained eigenvalue Problems (2) and (3) can be easily reduced to an equivalent unconstrained eigenvalue problem with the initial operator modified as follows (see [
1] for more details):
where
is the projection operator
.
The key moment of the asymptotic projection method is based on the following proposal [
5,
6,
35]:
The constraint vector tends to an eigenvector of the modified operator
Then, fulfillment of Equation (3) will follow automatically due to the orthogonality of the eigenvectors, which correspond to different eigenvalues of a self-conjugate operator. The only additional computation beyond that required for the ground state (see Equation (1)) is the evaluation of the overlap matrix element . This remark is important, especially, when different optimized basis sets are used for different states.
3. Matrix Kohn–Sham-Like Equations for Highly Excited States
In this Section we shall show how this methodology can be easily extended to high excitations, including doubly, triply and so on excitations. For example, a doubly excited state
Φdouble produced by excitation of electrons from
and
orbitals can be obtained by using the following orthogonality constraints imposed on orbitals {
ϕj} of the doubly excited state Slater determinant
Φdouble:
and
Then, using the asymptotic projection methodology, we have the corresponding KS-like equations in a form:
with the orthoprojectors:
and
, and the operator
with an effective potential
where
,
are the external and the Hartree (classical Coulomb) potentials and
is a local exchange-correlation potential.
In order to obtain equations for second excited state we should substitute the operators
and
by the orthoprojectors
where
and
are orbitals from the first ES determinant. In addition, in general, indices
k or
l for orbitals in different determinants may be different. It allows us to construct various classes of excited states in a unique approach. It is clear that for arbitrary
N-th excited state, we have
In practical applications, we invariably invoke the algebraic approximation by parameterizing the orbitals in a finite one-particle basis set. This approximation may be written
It should be stressed that, in general, the basis set for the excited state, is distinct from that for the ground state. This is because calculations on excited states can be meaningless without including sufficiently diffuse basis functions.
Once the basis set
is introduced, the integro-differential Equations (7) and (8) become generalized matrix eigenvalue problems:
with
is the overlap matrix and
are modified
h matrices with the elements
Since
cannot be infinity in practical calculations, one has to settle on some large finite values. Reasonable compromise is achieved if
is of order of 10
3–10
4 hartrees. This value provides a target accuracy for
and
~ 10
−5 [
35] and leads to orthogonality of the KS determinants which describe excited states.
After solving Equation (14) with Matrices (15) and (16) the excited state energy is determined by
where
ρ is the electron density and
is the exchange-correlation energy functional. If we use an exchange-only approximation then
should be replaced by the exchange energy functional
expressed in terms of orbitals from Equations (7) and (8):
In concluding this section, it is worth also noting that the imposition of the orthogonality constrainton an approximate lower state wave function, such as the Kohn–Sham reference function, does not, in general, yield an excited state energy which is an upper bound to the exact excited state energy.
4. Results of Calculations and Their Discussion
In this section we demonstrate the potential of our exchange-only COEP (x-COEP) implementation for highly excited states of the He atom (1sns, n = 2, 3,…, 7; 2sns, n = 3, 4,…, 8; 3sns, n = 4, 5, …, 8; and 4sns, n = 5, …, 8) and highly excited 1s2ns (n = 3, …, 7) doublet states of the Li atom.
Before discussing results of calculations, we should point out that our x-COEP implementation employs a parameterized form of
Veff proposed in [
36] and further developed in [
37,
38] where an effective potential is a direct mapping of the external potential
Vext and for
N-electron atoms takes the form:
In the spirit of the OEP method, variational parameters
C and
d are determined by minimizing the energy
for each individual state. The potential preserves symmetry properties of the exact eigenstates and has proven to be successful for the ground state calculations of different characteristic atoms and molecules [
34,
37,
38] and for low-lying excited states [
1,
2]. Therefore it is natural to try this potential for highly excited state calculations.
Our results are compared with the Hartree-Fock (HF) energies computed in Ref. [
39] and high precision data obtained by the configuration interaction method in the basis set of Hylleraas functions that explicitly depend on interelectron separations [
40,
41]. Calculations were carried out using 42
s-gaussians.
which were constructed according to the even-tempered prescription, i.e., the exponents,
ζp, were defined by the geometric series:
The parameters
α and
β 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. [
42].
We performed excited state calculations for the He atom using the different schemes:
- (i)
Common basis set adjusted to the ground state and the potential parameters optimized for a given excited state were employed (x-COEP-bgs column in
Table 1). We restricted parameters to 3 excited states to show some tendencies.
- (ii)
Excited states were obtained from the ground state Hamiltonian but basis sets were optimized for each individual excited state (x-COEP-Vgs column in
Table 1 and
Table 2).
- (iii)
Both orbital basis sets and the potential parameters were optimized for a given excited state (x-COEP column in these tables).
The corresponding results are listed for triplet singly excited 1
sns (
n = 2, 3, …, 7) states (
Table 1) and triplet doubly excited state energies of 2
sns (
n = 3, 4, …, 8) states (
Table 2). x-COEP energies of triplet singly excited energies were compared with the HF energies obtained with the maximum overlap method (MOM) [
43] which does not use orthogonality restrictions. The calculations in [
43] were carried out using 70
s even-tempered Slater-type basis functions. The results of [
43] can be considered as benchmark HF data. These authors used the extended precision in the “Mathematica” package to avoid problems with almost linearly dependent basis set. Unlike Ref. [
43] our calculations were restricted to 6 states (up to 1
s7
s) because for
n > 7 we observed that the corresponding optimal basis sets present some linear dependencies. We used double precision and controlled the conditionality of the overlap matrix by calculating its eigenvalues in order to avoid the appearance of computational linear dependence. Calculations of Ref. [
43] were restricted to only singly excited states. Therefore we compare doubly excited energies with our HF calculations [
39] and accurate theoretical calculations (named in
Table 2,
Table 3 and
Table 4 as
Eexact) based on a configuration interaction approach with the explicitly correlated Hylleraas basis set functions [
40]. A comparison of excited state energies presented in x-COEP-bgs and x-COEP-Vgs columns of
Table 1 with the fully optimized results (x-COEP column) shows that basis set optimization plays a crucial role for a correct description of excited states with respect to optimization of potential parameters. We observed that potential parameters for the ground state differ from those for excited states. However, we did not notice any trends in their behavior. In the case of helium the dependence of the results on the values of the parameters is relatively weak and the same effective potential correctly describes both the ground state and the excited states. The columns x-COEP-Vgs in
Table 1 and
Table 2 demonstrate that the ground state potential with parameters
C = 3.982687 and
d = 0.248872 and the optimization of the orbital basis for a given excited state can support a reasonable accuracy of excited energies.
In
Table 3 and
Table 4 triplet doubly excited energies of 3
sns (
n = 4, 5, …, 8) states and 4
sns (
n = 5, …, 8) of He, computed at the x-COEP level, are given. One can see that the accuracy of the x-COEP calculations is improved when
n increases. This observation is in agreement with Ref. [
43] whose authors pointed out that “In those states where
n >> 1, the electrons are spatially well separated and one might anticipate intuitively that they will be weakly correlated and that the x-COEP and Hartree-Fock methods, which neglects such effects, may be an excellent approximation”.
Our observations of
Table 2,
Table 3 and
Table 4 show that energy differences
ECOEP −
Еexact for different excited states are similar. As a result, excitation energies based on the x-COEP method are in good agreement with those computed with highly correlated methods (see
Table 5).
In
Table 6 we compare x-COEP excited doublet 1
s2ns (
n = 3,…, 7) energies and excitation energies of the Li atom to the HF [
38] and “exact” energies obtained with the most accurate configuration interaction wave function using the Hylleraas basis set [
41]. It is known that high accuracy of calculation of the transition frequencies for Rydberg states of alkali metal atoms, in particular lithium, is a topical problem of theoretical methods for studying the electronic structure [
44]. Such calculations for highly excited states have been carried out only recently [
41]. Comparison of the calculated energies of excitation from the 1
s23
s state with the high precision results [
39] shows that the proposed implementation for ESs yields excellent agreement with the precision excitation energies (compare columns 5 and 6 of
Table 6). Our calculation of the x-COEP ground state (1
s22
s) energy yields
E = −7.431724 hartrees. Comparison with the precision result
Eexact = −7.478060 hartrees gives
E(xCOEP) –
Eexact = 0.046336 hartrees, which is inagreement with the accuracy of ES energy determination. For example, for the 1
s26
s state
E(xCOEP) –
Eexact = 0.046655 hartrees. For the Li atom we collected also the optimum values of parameters “
C” and “
d” defining the effective potential (see
Table 7). We can see that, unlike He, there aresome trends in their behavior: the “
C” parameter increases for higher excited states whereas the “
d” parameter decreases.
Thus, results show that the x-COEP implementation of TI-DFT can ensure the accuracy of energy determination for highly excited states comparable in accuracy to the ground state and can be considered to be the starting point for the development of many-body methods of calculation of correlation effects for such states. The computational cost of the method for highly excited states requires virtually the same computational efforts as for the ground state.