## Chapter 7. Orbital-Dependent Functionals and Exact-Exchange Methods.

Development note Development of this chapter has been temporarily delayed until the chapters covering the basics of molecular spectroscopy have been completed.

## 7.1. Introduction to very accurate density-functional methods.

As seen from our examples of density-functional theory applied to the helium atom and the hydrogen molecule, alternative approaches are necessary to improve the accuracy of the Kohn-Sham orbital energies. In particular, none of the first or second generation functionals , such as the LDA or the generalized gradient approximation (GGA) give accurate Kohn-Sham orbitals or eigenvalues. Because the exchange-correlation functionals are approximations in what is in principle an exact theory of electronic structure, the resulting Kohn-Sham eigenvalues can differ significantly from what one might expect if no approximation to the functionals was required . A brief summary of some of the deficiencies are described in [Koch,p.88], most of which can be understood by comparing the exchange-correlation potential

1. The asymptotic behavior of the potential should behave as  ${v}_{\mathrm{xc}}\to -\frac{1}{r}$   as $r\to \infty$  to cancel the Hartree self-interaction terms (e.g.  ${v}_{x}=-{v}_{H}/2$ in the case of two-electron systems). Many of the common functionals yield a potential  ${v}_{\mathrm{xc}}$  with a spurious divergence at the nucleus and does not give the correct -1/r long-distance behavior [Koch,p.88].

2. ${v}_{\mathrm{xc}}$  should be self-interaction free, which can be expressed as follows    

and

3. The energy eigenvalue of the highest occupied Kohn-Sham orbital  ${\epsilon }_{N}$  should be exactly equal to negative of the ionization energy,  ${I}_{N}=-{\epsilon }_{N}={E}_{o}^{N-1}-{E}_{o}^{N}$,  where  ${E}_{o}^{N}$  is the ground-state energy for a N-electron system .

4. For a closed subshell system,  ${v}_{\mathrm{xc}}$  is a discontinuous function of the number of electrons N.  ${v}_{\mathrm{xc}}$  must jump by the amount  ${I}_{N}-{A}_{N}$  as the number of electrons N is increased from N to N-δ, where δ is an infinitesimally small positive constant, and  ${A}_{N}={E}_{o}^{N}-{E}_{o}^{N+1}$  is the electron affinity .

Satisfying these conditions is critical in obtaining accurate orbitals and eigenvalues, and it is also an important first step in extending density-functional theory to describe many different physical properties involving electronic structure. In particular, the difference between the Kohn-Sham orbital energies provides a first approximation to the optical excitation energies.

Several different methods have been developed to obtain exact functionals and exchange-correlation potentials  ${v}_{\mathrm{xc}}$. Umrigar and Gonze did one of the first studies using exact Kohn-Sham potentials in their work on helium and two-electron ions . They started with very accurate wave functions, obtained by other methods, and then numerically integrated these to obtain nearly exact two-electron densities. Using their "almost exact" two-electron densities they calculated the exchange and correlation potentials, energy densities, total energies, and compared the results to those using various LDA and GGA functionals, also calculated using their exact two-electron densities. In this way, they were able to identify the shortcomings of various approximate functionals. Others have used the exact two-electron potentials  ${v}_{\mathrm{xc}}$  of Umrigar and Gonze to isolate and identify deficiencies in various functionals used in time-dependent density-functional theory (TDDFT)  .

Both TDDFT    and DFT perturbation theory  are methods that provide a formal connection between the Kohn-Sham orbital energies and excited states. For both methods the difference between Kohn-Sham orbital energies are the zeroth order contribution to the excitation energies, i.e.

Savin, Umrigar and Gonze  showed that if a very accurate exchange-correlation potential is used, then the difference between the Kohn-Sham orbital energies,  ${\mathrm{\Delta \epsilon }}_{\mathrm{KS}}$,  provides a good first approximation to the excitation energies , as can be seen in Table 1 in the case of the helium atom. In fact, for the atoms studied by Filippi et al. , the Kohn-Sham orbital energies always lie in between the exact singlet and triplet energy levels, with the one exception being the  $1s\to 3d$  transition in ionized lithium.

In contrast to these exact methods, the LDA or the generalized gradient approximation (GGA), in many cases, will not yield any unoccupied bound states (i.e.  ${\epsilon }_{k}>0\text{\hspace{0.17em}for\hspace{0.17em}}k>{N}_{\sigma }$), in which case Eq. (7.4) would not be valid. For the helium atom and the hydrogen molecule, this characteristic of the unoccupied (virtual) orbitals produced by the LDA and GGA can be seen from the output of the first example program, freeMQM-1.cpp. The orbital energies are stored in the array Diag[I] and the energy of the lowest energy virtual orbital is stored in the second element of the array, Diag. For both the helium atom and the hydrogen molecule, the eigenvalue of even the lowest unoccupied orbital is greater than zero,

What is more, in the case of helium, the LDA and the GGA give excitation energies that are greater than the ionization energy, i.e.

These kinds of deficiencies highlight the importance of obtaining more accurate approximations than the LDA and the GGA.

TransitionFinal stateExperimentDrakeΔεKSΔεKS + ΔE(1)
1s $\to$ 2s23S0.728330.728500.74600.7232
21S0.757590.75775 0.7687
1s $\to$ 2p13P0.770390.770560.77720.7693
11P0.779720.77988 0.7850
1s $\to$ 3s33S0.834860.835040.83920.8337
31S0.842280.84245 0.8448
1s $\to$ 3p23P0.845470.845640.84760.8453
21P0.848410.84858 0.8500
1s $\to$ 3d13D0.847920.848090.84810.8481
11D0.847930.84809 0.8482
1s $\to$ 4s43S0.867040.867210.86880.8667
41S0.869970.87014 0.8710

The last column of Table 1 shows the result of including the first order shift from Görling-Levy perturbation theory (GLPT) as calculated by Filipi et al.  using their nearly exact exchange-correlation potentials. It can be seen that the first order correction  ${\mathrm{\Delta E}}^{\left(1\right)}$  shifts singlet and triplet energies toward the experimental values.

## 7.4. Görling-Levy perturbation theory (GLPT).

In a future revsion of this document, we will see that in the GLPT approach, an adiabatic connection parameter α is treated as the expansion parameter in perturbation theory . We will also see in TDDFT, that the Kohn-Sham levels are split and shifted toward their true values by contributions from the time-dependent xc kernel in the linear-response regime, where the xc kernel is ...to be continued.

## 7.5. The Optimized Effective Potential Method (OEP).

In a future revision of this document, we will also show how the optimized effective potential method provides a procedure to transform the nonlocal Hartree-Fock (HF) exchange operator into a local variationally optimized exchange potential that can be incorporated into Kohn-Sham density-functional calculations. Sharp and Horton  and Talman and Shadwick  used variational principles   in the original development of the OEP integrodifferential equations. Other methods have been used to derive the OEP equations which do not require any discussion of the optimized exchange potential method itself . Görling and Levy, for example, have derived the OEP equations using perturbation theory  and Shaginyan used linear-response theory . It was explained by Sahni, Gruenebaum, and Perdew  and Perdew and Norman  that the Talman-Shadwick energy-optimized local potential is “an exact realization of the Kohn-Sham potential for ‘exchange only’ in atoms” and that the only difference between the exchange-only Kohn-Sham and Hartree-Fock theories is that the Hartree-Fock effective potential is nonlocal.

Application of the OEP in its original form is computationally demanding, which has limited its application to a few simple systems . Krieger, Li, and Iarafe, however, presented an approximation that helped revitalize interest in the method  . Kim et al.  have implemented the Krieger-Li-Iafrate (KLI) approximation to the optimized effective potential (OEP) exact exchange functional to study small molecules, using a three-dimensional (3D) finite-difference pseudopotential method  . They found that the addition of LDA or GGA correlation functions did not lead to systematic improvements over the KLI exchange-only approximation. It has been shown by Vasilev et al. , however, and further explained by Burke et al. , that in TDDFT the correlation effects are more important than in calculations of the time-independent ground state properties. In particular, an antiparallel spin correlation functional  ${f}_{\mathrm{xc}}^{\mathrm{↑↓}}$  contributes to the splitting of the Kohn-Sham energy eigenvalues . ...to be continued.