Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367

www.elsevier.nl/locate/theochem

Molecular excitation energies from time-dependent density

functional theory q
T. Grabo, M. Petersilka, E.K.U. Gross*
Institut für Theoretische Physik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

Abstract
The performance of various exchange-correlation functionals is evaluated in the calculation of molecular excitation energies
from time-dependent density functional theory. Excitation energies of N2 and CO are reported, using either the local density
approximation (LDA) for exchange and correlation or an orbital functional in the approximation of Krieger, Li and Iafrate. The
latter is based on exact exchange plus a correlation contribution in the form suggested by Colle and Salvetti. While the LDA
proves to work remarkably well for the lower excited states due to error cancellations, self-interaction-free potentials are
essential for a good description of higher lying states. q 2000 Elsevier Science B.V. All rights reserved.
Keywords: Molecular excitation energies; Time-dependent density functional theory; Local density approximation

1. Introduction and dedication interaction calculations [6]. In the present paper, we

will investigate the performance or this optimized
It is a great pleasure for us to contribute to this effective potential in the calculation of molecular
scientific celebration of Professor Rezso Gáspár. excitation energies.
With his suggestion of a local exchange potential, As far as the determination of excitation spectra is
Gáspár was among the pioneers of a theory which, concerned, several extensions of ground-state DFT
since the work of Hohenberg, Kohn and Sham [1,2], have been proposed. They are based either on the
has been termed density-functional theory (DFT). Rayleigh–Ritz principle for the lowest eigenstate of
Over the years, the available approximations for the a given symmetry class [7–9] or on a variational prin-
local exchange potential have steadily improved [3– ciple for ensembles [10–23]. A major difficulty lies in
5]. The variationally best local exchange potential is the fact that the exchange-correlation (xc) functionals
obtained if the Hartree–Fock total energy is varied associated with these approaches are not identical
under the subsidiary condition that the orbitals come with the ordinary ground-state xc energy functional,
from a local potential. The resulting so-called opti- although the latter sometimes gives rather accurate
mized effective potential, when complemented with results [23]. In principle, the xc functional in these
a suitable correlation contribution, was recently approaches should depend either on the symmetry
shown to yield atomic ground-state properties in labels of the prescribed symmetry class or on the
close agreement with results based on configuration particular ensemble considered, and very little is
known about the nature of this dependence. To
q circumvent this problem, we recently proposed
Dedicated to Professor R. Gáspár on the occasion of his 80th
year. [24–28] a different approach for the calculation of
* Corresponding author. excitation energies, which is based on time-dependent
0166-1280/00/$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S0166-128 0(99)00445-5
354 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367

density-functional theory (TDDFT) [29]. This method electrons with spin s subject to a frequency-depen-
makes use of the fact that excitation energies may be dent perturbation V1s 0 …r 0 ; v† has poles at the exact
obtained from the poles of the linear density response excitation energies (see Ref. [42]). In terms of the
function which can be expressed, in principle exactly, density–density response function xs;s 0 of interacting
in terms of the response function of a noninteracting electrons, the frequency-dependent density response
Kohn–Sham (KS) system and a frequency-dependent r 1s (r,v ) is given by
exchange-correlation kernel. In this way, corrections
to the KS orbital energy differences (which are the XZ
r1s …r; v† ˆ d3 r 0 xs;s 0 …r; r 0 ; v†V1s 0 …r 0 ; v†: …1†
poles of the KS response function) are obtained, s0
which shift them towards the true excitation energies
of the fully interacting system. As input, the resulting Alternatively, the time-dependent generalization of
computational scheme only requires ground state DFT allows one to express the exact density response
properties, i.e. occupied and virtual orbitals of the r 1s via the response function xss;s 0 of the noninteract-
KS potential corresponding to the ground-state of a ing KS system [27]:
given physical system. Recent applications to atoms
[27,28,30–32], molecules [33–38] and clusters [39– XZ
41] are highly promising. In particular, the successful r1s …r; v† ˆ d3 r 0 xsss 0 …r; r 0 ; v†Vs1s 0 …r 0 ; v†: …2†
s0
calculation of the excited-state potential energy
surfaces of formaldehyde [38] has shown that In the above equation, Vs1s 0 is the linearized time-
TDDFT is also capable of describing the strong dependent KS potential given by
mixing with Rydberg transitions and the correspond-
ing avoided crossings. In view of these successes, one Vs1s 0 …r 0 ; v†ˆV1s 0 …r 0 ; v†
can expect that this scheme will become a standard
method for the calculation of excitation energies of X Z 3 00  1 0 00

1 dr 1 f 0
xcs s 00 …r ; r ; v†
finite many-particle systems.
s 00
ur 0 2 r 00 u
The accuracy of any density functional method,
however, crucially depends on the quality of the  r1s 00 …r 00 ; v†: …3†
functional approximations involved. The purpose of
this work is to investigate the role of the static KS The spin-dependent xc kernel fxc is defined as the
potential from which the orbitals and orbital energies Fourier transform of
entering the scheme are calculated. Furthermore,

we address the influence of the necessary truncation dVxcs ‰r" ; r# Š…r; t† 
of the matrix equation from which the corrections fxcss 0 …r; t; r 0 ; t 0 † :ˆ  …4†
drs 0 …r 0 ; t 0 † rGS" ;rGS#
to the KS orbital energy differences have to be
determined.
evaluated at the ground-state spin densities rGS" ; rGS#
This article is organized as follows: In Section 2, a
of the unperturbed system. The response function of
brief review of the underlying theory is given. In
the KS system can be expressed in terms of the
Section 3, the employed approximate functionals are
stationary KS-orbitals
described. Section 4 discusses the numerical results
obtained for diatomic molecules, followed by a
xsss 0 …r; r 0 ; v†
summary in Section 5.
X wjs …r†wpks …r†wpjs …r 0 †wks …r 0 †
ˆ dss 0 … fk s 2 f j s † ;
j;k
v 2 …ej s 2 ek s † 1 i h
2. Basic formalism
…5†
The calculation of excitation energies from time-
dependent DFT makes use of the fact that the full where fis denote the Fermi-occupation factors (1 or 0).
linear density response r 1s of a system of interacting With these definitions, Eq. (2) may be rewritten as
 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367 355

an integral equation for the linear density response: The corresponding transition energies are
" vqs ˆ ejs 2 eks : …10†
X Z 3 00 XZ 3 0
d r dss 00 d…r 2 r 00 † 2 d r xsss 0 …r; r 0 ; v†
s 00 s0
Moreover, we define
!# Fqs …r† U fks …r†p fjs …r†; …11†
1
 1 fxcs 0 s 00 …r 0 ; r 00 ; v† 00
r1s 00 …r ; v†
ur 0 2 r 00 u aqs :ˆ fks 2 fjs ; …12†
XZ
ˆ d3 r 0 xsss 0 …r; r 0 ; v†V1s 0 …r 0 ; v†: (6) and set
s0 XZ Z
jqs …v† :ˆ d3 r 0 d3 r 00 Fqs …r 0 †p
s 00
In general, the true excitation energies V are not iden-
 
tical with the Kohn–Sham excitation energies ejs 2 1
 1 fxcss 00 …r ; r ; v† gs 00 …r 00 ; v†:
0 00
eks : Therefore, the right-hand side of Eq. (6) remains ur 0 2 r 00 u
finite for v ! V:. Since, on the other hand, the exact …13†
spin-density response r 1s , has poles at the true excita-
tion energies V , the integral operator acting on r 1s on Using these definitions, Eq. (7) can be recast into
the left-hand side of Eq. (6) cannot be invertible for X aqs Fqs …r†
v ! V: (Assuming the existence of the inverse opera- j …v† ˆ l…v†gs …r; v†: …14†
q v 2 vqs 1 ih qs
tor, its action on both sides of Eq. (6) results in a finite
right-hand side for v ! V: This leads to a contradic- Solving this equation for g s (r,v ) and reinserting
tion since r 1s , remaining on the left-hand side, has a the result on the right-hand side of Eq. (13) we
pole at v ˆ V.) arrive at
Consequently, the true excitation energies V are X X Mqsq 0 s 0 …v†
characterized as those frequencies where the eigen- j 0 0 …v† ˆ l…v†jqs …v†; …15†
values of the integral operator acting on the spin- s0 q0
v 2 vq 0 s 0 1 ih q s
density vector in Eq. (6) vanish. Integrating out the
delta-function in Eq. (6), the true excitation energies where we have introduced the matrix elements
V are those frequencies, where the eigenvalues l (v ) of Z Z
Mqsq 0 s 0 …v† ˆ aq 0 s 0 d3 r d3 r 0 Fpqs …r†
XZ XZ
d3 r 0 xsss 0 …r; r 0 ; v† d3 r 00  
1 0
s0 s 00  1 f xcss 0 …r; r ; v† Fq 0 s 0 …r 0 †:
! ur 2 r 0 u
1 …16†
× 1 fxcs 0 s 00 …r ; r ; v† gs 00 …r 00 ; v†
0 00
ur 0 2 r 00 u
Introducing the quantity bqs U jqs …V†=…V 2 vqs †
and using the condition (8), we can, at the corre-
ˆ l…v†gs …r; v† (7)
lated excitation energies v ˆ V; rewrite Eq. (15)
in the following form:
satisfy XX
…Mqsq 0 s 0 …V† 1 vqs dqq 0 dss 0 †bq 0 s 0 ˆ Vbqs : …17†
l…V† ˆ 1: …8† s0 q0

This condition rigorously determines the true excitation Once again, this eigenvalue problem rigorously
spectrum of the interacting system at hand. determines the true excitation spectrum of the
For a single-particle transition …k ! j† we introduce interacting system.
the notation In practice, the matrix equation (15) or, alter-
natively, the eigenvalue problem (17), has to be trun-
q ; … j; k†: …9† cated in one way or another. One possibility consists
356 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367

in expanding all quantities in Eq. (15) about one large basis sets stability and convergence problems
particular KS-orbital energy difference 1 v pt . For have been reported [33]. Results from fully numerical
non-degenerate poles one obtains [27] in lowest order codes that solve the KS equations on a grid have the
advantage of being free of errors caused by the finite
V ˆ vpt 1 Mptpt …vpt †: …18†
size of the basis set. The disadvantage is that one can
If the pole v pt is P-fold degenerate, i.e. consider only bound orbitals, as states with positive
energy are not represented accurately.
vp1 t1 ˆ vp2 t2 ˆ … ˆ vpPtP ; v0 ; …19†

then, in lowest order, the corresponding excitation 3. Approximate functionals

energies V n are given by [28]
Vn ˆ v0 1 Re …An …v0 †† …20† Apart from the truncation of the matrix equations
(15) or (17) described in Section 2, two further
where the An(v 0) are the P eigenvalues of the trun- approximations are necessary: (i) in the calculation
cated matrix equation of the KS orbitals f k(r) and their eigenvalues e k,
X
P one employs some approximation of the static xc
Mpi ti pk tk …v0 †jp…n†
k tk
…v0 † ˆ An …v0 †jp…n†
i ti
; potential Vxc. (ii) The functional form of the dynamic
kˆ1 …21† xc kernel fxc needs to be approximated.
In this work, two approximations of the exchange-
i ˆ 1; …; P correlation potential are used: (i) the LDA in the para-
This lowest-order result amounts to approximating the meterization of Vosko, Wilk and Nusair [44] and (ii)
KS response function x s by the single pole contri- orbital dependent xc functionals
bution at v 0 alone. It therefore will be referred to as Exc ‰{wis }Š ˆ Exexact ‰{wis }Š 1 Ec ‰{wis }Š …23†
single-pole approximation (SPA). The resulting
excitation energies can be assigned to symmetry including the exact exchange energy expression
labels according to the symmetry of the matrix M in
1 X X Ns
Eq. (21). Exexact ‰{wis ‰rŠ}Š ˆ 2
Alternatively, Eq. (17) may be truncated by only 2 s ˆ";# j;kˆ1
considering the matrix elements corresponding to a …24†
Z Z wpjs …r†wpks …r 0 †wks …r†wjs …r 0 †
particular single-particle transition …k ! j† and the
 respec- × d3 r d 3 r 0 :
reverse transition …j ! k†; denoted by Q and Q; ur 2 r 0 u
tively. This leads to the following eigenvalue
problem: The corresponding xc potential is evaluated using
X X the semi-analytical method of Krieger, Li and Iafrate
…Mqsq 0 s 0 …V† 1 vqs dqq 0 dss 0 †bq 0 s 0 ˆ Vbqs : (KLI) [45–51] given by
s 0 q 0 ˆQ;Q
1 X Ns
…22† Vxcs …r† < uw …r†u2 ‰uxcis …r† 1 …V KLI xcis †Š
xcis 2 u
rs …r† iˆ1 is
We will refer to this scheme as small matrix approx-
…25†
imation (SMA).
In the framework of traditional quantum chemistry, with
Eq. (17) is usually solved by expanding the orbitals
1 dExc ‰wjs Š
and potentials in a basis set. This has the advantage uxcis …r† U : …26†
that a proper choice of the basis can lead to a good wpis …r† dwis …r†
representation of the continuum contributions (see, The constants …V KLI
xcis 2 u  xcis † denote average values
for example Ref. [43]). However, even with very taken over the density of the is orbital, i.e.
Z
1
This is justified if the true excitation energy V is not too far from
uxcis ˆ d3 ruwis …r†u2 uxcis …r† …27†
the KS orbital energy difference v pt .
 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367 357

Table 1 and
Orbital energies for N2 at R ˆ 2:0744 a:u: from various DFT 
approaches axc …r† 
GALDA …r; r 0 ; v† 0
ˆ d…r 2 r † 2  …29†
xcLDA xcLDA a KLI KLICS
xc
m0 r rGS …r†

ETOT 2108.6999 2108.6957 2108.9852 2109.4629 Here, axc …r† ˆ …22 =2z2 † …ehom
xc …r; z††uzˆ0 is the
Occupied orbitals exchange-correlation contribution to the spin stiffness
1sg 213.9666 213.9677 214.3728 214.4126 of the homogeneous electron gas with relative spin-
1su 213.9652 213.9662 214.3715 214.4113 polarization z ˆ 0; and m 0 denotes the Bohr mag-
2sg 21.0379 21.0411 21.3061 21.3384
2su 20.4938 20.4939 20.7457 20.7754
neton. In our calculations, the parameterizations of
1pu 20.4370 20.4366 20.6809 20.7131 Ref. [44] were used for e xc and a xc. The diagonal
3sg 20.3826 20.3840 20.6303 20.6605 elements of the spin dependent exchange-correlation
Unoccupied orbitals kernel are given by
1pg 20.0813 20.0816 20.3100 20.3412
4sg 20.0015 0.0000 20.1854 20.1988 fxc"" ˆ fxc## ˆ fxc 1 m20 Gxc ; …30†
2pu .0 0.0235 20.1486 20.1582
3su .0 0.0114 20.1300 20.1390 and the off-diagonal elements read
1dg .0 – 20.1016 20.1081
5sg .0 0.0305 20.0914 20.0986 fxc#" ˆ fxc"# ˆ fxc 2 m20 Gxc : …31†
6sg .0 – 20.0855 20.0886
2pg .0 0.0481 20.0799 20.0832 In the adiabatic approximation, the Fourier transforms
3pu .0 – 20.0749 20.0777 of the time-dependent kernels have no frequency
a dependence at all. Their value corresponds to the
Results obtained with a code using a basis set of 106 contracted
Gaussian-type orbitals from Ref. [33]. static (v ˆ 0) limit of the linear response kernels
[58]. The frequency dependence however, still enters
the scheme via the pole-structure of the non-interact-
and likewise for V KLI
xcis : In the following, we refer to
ing response-function x s.
this scheme as KLI if the exact exchange energy func- It is important to realize that, although the formal-
tional (24) is used with correlation contributions ism makes use of the time-dependent generalization
neglected. By KLICS we denote the same method of DFT, only ground-state properties are required in
with the inclusion of correlation contributions in the the actual calculation.
form given by Colle and Salvetti [6,52,53,54].
It is a well-known fact that the LDA xc potential 4. Numerical results for molecular systems
falls off exponentially. Consequently, the LDA
valence-orbitals are too weakly bound, with orbital For calculations on diatomic molecular systems, we
energies being in error by up to 100%. Contrary used a fully numerical, basis-set free two-dimensional
to the LDA, both the KLI and KLICS potential code [54,57], developed from the Xa program written
show the correct 21=r tail for large r [55] (for a by Laaksonen, Sundholm and Pyykkö [59–61]. The
recent review article see Ref. [56]). This leads to code solves the one-particle Schrödinger equation for
orbital energies very close to the exact ones diatomic molecules
[6,49,57]. !
Finally, the frequency-dependent quantity fxcss 0 72 Z1 Z2
2 2 2 1 VH …r† 1 Vxs …r†
KLI
has to be approximated. In the present work 2 uR1 2 ru uR2 2 ru
we will restrict ourselves to the use of the adia-
batic LDA (ALDA) which can be expressed via the  wjs …r† ˆ ejs wjs …r†; (32)
quantities
where Ri denotes the location and Zi the nuclear

 charge of the ith nucleus in the molecule. The partial
0 2
2
0 
fxc …r; r ; v† ˆ d…r 2 r † 2 …rexc …r††
ALDA hom
…28† differential equation is solved in prolate spheroidal
2r r
GS …r† coordinates on a two-dimensional mesh by a
358 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367

Table 2
Excitation energies for N2 from an xcLDA-calculation at R ˆ 2:0744 a:u: The LDA was employed for Vxc and the ALDA for the xc kernels.
Dv KS denotes the KS orbital energy difference. All numbers in mHartrees

State Dv KS SPA SMA Full a Full b Exp c

a 1Pg 3sg ! 1pg 301.4 344.3 341.6 339.4 334.4 342.1

B 3Pg 281.0 280.2 280.1 279.3 295.5
a 0 1 Su 2 1pu ! 1pg 355.8 355.8 355.8 355.8 355.0 364.6
B 0 3 Su 2 355.8 355.8 355.8 355.0 355.4
A 3 Su 1 301.6 296.7 296.7 289.6 284.5
w 1Du 378.3 377.6 377.6 375.6 377.4
W 3Du 328.7 327.6 327.6 324.5 326.3
a 00 1 Sg 1 3sg ! 4sg 381.1 385.3 385.3 385.3 – 448.3
E 3 Sg 1 379.9 379.8 379.8 – 441.0
o 1Pu 2su ! 1pg 412.5 521.3 509.8 509.8 – 500.9
C 3Pu 384.9 383.9 383.7 380.7 411.2
c 1Pu 1pu ! 4sg 435.5 435.4 435.3 435.3 – 474.1
3
Pu 434.8 434.8 434.9 – 470.3
a
Neglecting continuum states.
b
Basis-set calculation including continuum states from Ref. [33].
c
From Ref. [43].

relaxation method, while the third variable, the from Ref. [33] of an LDA calculation using an expan-
azimuthal angle, is treated analytically. The Hartree sion of the orbitals and potentials into a large
potential contracted Gaussian-type basis are displayed.
Comparison with our numerical results in the first
Z r…r 0 † column shows that the basis-set-expansion error is
VH …r† ˆ d3 r 0 …33†
ur 2 r 0 u fairly small: the total energy is 4.2 mHartrees too
high and the orbital energies show an error of the
and the functions uxis (r) (cf. Eq. (26)) needed for the same magnitude. For the calculation of excitation
calculation of the exchange potential VxKLIs …r† (cf. Eq. energies, the orbital-energy differences are important.
(25)) are computed as solutions of a Poisson and They show an error of about 1 mHartree on average.
Poisson-like equation, respectively. In this step, the Furthermore, the unbound orbitals with positive ener-
same relaxation technique as for the solution of the gies show a different energetic order as the corre-
one-particle Schrödinger equation is employed. A sponding ones from the KLI and KLICS
very detailed description of the code is given in Ref. calculations. Again, this indicates the poor quality of
[62]. the LDA potential in the asymptotic region. In
We have applied our method to the nitrogen and addition, this fact is also responsible for the poor qual-
carbon monoxide molecules. Both are well-studied ity of the value for e HOMO, which should be equal to
systems for which most of the lower excited states the first ionization potential in exact DFT. In LDA,
have been observed and measured in detail. eHOMO ˆ 2383 mHartrees; whereas the experimental
In Table 1, we show the ground-state and orbital ionization potential is 2573 mHartrees [33]. The KLI
energies of N2 obtained with the LDA, KLI and value for e HOMO is 2630 mHartrees and thus signif-
KLICS potentials. Owing to the wrong behaviour of icantly closer to the exact one. However, adding the
the LDA potential in the asymptotic region, there are correlation contribution of Colle and Salvetti to the
only two bound unoccupied orbitals in this approxi- KLI scheme results in a value which is further away
mation. For the KLI and KLICS approaches, leading from the exact one than the one obtained within the
to the correct 21=r tail of the xc potential, we show KLI approach. This indicates that the CS correlation
the lowest nine unoccupied orbitals, all of which have potential needs some improvement when calculating
negative energies. In the second column, the results molecular properties.
 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367 359

Table 3 Table 4
Excitation energies for N2 from a KLI-calculation at Excitation energies for N2 from a KLI-calculation at
R ˆ 2:0744 a.u. The exact exchange functional was employed for R ˆ 2:0744 a.u. The CS approximation for correlation added to
Vxc in KLI-approximation and the ALDA for the xc kernals. the exact exchange was employed for Vxc in KLI-approximation
Dv KS denotes the KS orbital energy difference. All numbers in and the ALDA for the xc kernals. Dv KS denotes the KS orbital
mHartrees energy difference. All numbers in mHartrees

State Dv KS SPA SMA Full a Exp b State Dv KS SPA SMA Full a Exp b

a 1Pg 3sg ! 1pg 320.2 364.1 361.4 358.7 342.1 a 1Pg 3sg ! 1pg 319.3 363.6 360.9 358.1 342.1
B 3Pg 299.6 298.9 298.7 295.5 B 3Pg 298.5 297.7 297.6 295.5
a 0 1 Su 2 1pu ! 1pg 370.9 370.9 370.9 370.9 364.6 a 0 1 Su 2 1pu ! 1pg 371.9 371.9 371.9 371.9 364.6
B 0 3 Su 2 370.9 370.9 370.9 355.4 B 0 3 Su 2 371.9 371.9 371.9 355.4
A 3 Su 1 317.9 313.4 312.8 284.5 A 3 Su 1 318.2 313.6 312.9 284.5
w 1Du 392.7 392.1 392.0 377.4 w 1D u 394.2 393.5 393.4 377.4
W 3Du 344.4 343.4 343.2 326.3 W 3Du 345.1 344.0 343.8 326.3
o 1Pu 2su ! 1pg 435.7 545.3 534.2 526.2 500.9 o 1Pu 2su ! 1pg 434.2 544.8 533.5 525.9 500.9
C 3P u 407.7 406.8 406.4 411.2 C 3Pu 406.0 405.0 404.7 411.2
a 00 1 Sg 1 3sg ! 4sg 444.8 461.5 461.2 458.1 448.3 a 00 1 Sg 1 3sg ! 4sg 461.7 479.8 479.5 476.3 448.3
E 3 Sg 1 440.0 440.0 439.7 441.0 E 3 Sg 1 456.4 456.3 456.1 441.0
c 1Pu 3sg ! 2pu 481.6 480.5 480.5 480.3 474.1 c 1Pu 3sg ! 2pu 502.3 501.3 501.3 500.8 474.1
3
Pu 479.9 479.9 479.6 470.3 3
Pu 500.3 500.3 499.9 470.3
b 1Pu 1pu ! 4sg 495.5 496.1 496.1 496.1 486.6 b 1Pu 1pu ! 4sg 514.3 515.3 515.3 515.3 486.6
3
Pu 491.8 491.8 492.1 497.8 3
Pu 1 510.2 510.2 510.6 497.8
c 0 1 Su 1 3sg ! 4su 500.3 500.8 500.8 498.8 477.0 c 0 1 Su 3sg ! 3su 521.5 522.4 522.4 519.9 477.0
3
Su 1 496.9 496.9 496.8 466.9 3
Su 1 517.6 517.6 517.4 466.9
a a
Using all occupied and the lowest 9 unoccupied orbitals. Using all occupied and the lowest 9 unoccupied orbitals.
b b
From Ref. [43]. From [43].

In Table 2 results are given for the vertical particular transition, it comes as no surprise that an
excitation energies of N2, calculated from the expansion of the exact equation (15) around a single
LDA xc-potential. Apart from the molecular KS pole introduces a larger error than for states where
state, the orbital transition and the orbital-energy this difference is smaller. The SMA performs clearly
difference, the excitation energies are shown for better, showing only deviations of a few tenths of a
different truncations of the exact matrix equation mHartree from the solution of the full matrix equa-
(15) as discussed in Section 3. With our numerical tion. This is most drastic for the o 1Pu excited state,
code, we have taken into account all occupied states where the error of the SPA is strongly reduced by the
for the solution of the “full” matrix Eq. (17). For SMA. As far as the deviations between the “full”
comparison, we also show results which include solutions from the numerical code on one hand and
continuum contributions from a calculation using the basis-set code on the other hand are concerned,
the same basis set as the one used for the results there are two sources of error: the error introduced by
shown in Table 1, taken from Ref. [33]. The method the basis-set expansion and, in the case of the grid
used in these calculations is identical to the one solution, the error resulting from the neglect of the
outlined in Section 3. Finally, the last column displays continuum orbitals. As the differences are fairly
experimental values taken from [43]. It is evident small, being 6.1 mHartrees at the most (for the A
3 1
from the table, that the SPA (Eq. (21)) gives results Su state), and as the errors caused by the use of
in good agreement with the ones from the solution of basis set are of comparable magnitude, we conclude
the full matrix equation. The deviation is a few that the effect of the continuum contributions is of
mHartrees, being largest for the o 1pu state with minor importance, at least for the lower excitation
11.5 mHartrees. As the KS orbital energy difference is energies studied here. Finally, we note that the agree-
quite far from the experimental value for this ment of the calculated excitation energies with the
360 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367

Fig. 1. The singlet–triplet splitting of the 2su ! 1pg KS transition in N2 from the xcLDA and KLI calculations.

experimental values is very good for the lower states all 15 orbitals listed in Table 1. As far as the perfor-
involving transitions to the 1Pg orbital. From Table 1 mance of the SPA and SMA is concerned, a picture
it is evident that the LDA-KS orbital-energy differ- similar to the LDA results is found: both approxim-
ences for these transitions are close to the (nearly ations work very well, except that the SPA shows
exact) ones from the KLI potential. This indicates larger errors when the KS orbital-energy difference
that a cancellation of the LDA self-interaction errors is further away from the experimental value, as is
occurs, leading to the excellent results for the lower the case for the o 1Pu state. Comparison with the
excitation energies. Values corresponding to transi- experimental values shows that the agreement with
tions to the 4sg orbital, however, underestimate the the calculated ones is very good. In contrast to the
experimental values considerably, reflecting the fact LDA results, this is also true for the higher excitations
that the error cancellation for the KS orbital energy involving transitions to other than the 1pg orbital.
differences ceases to work so well. Generally, the calculated values are higher in energy
In order to study the effect of the KS potential on than the ones from the LDA approach. This is primar-
the excitation energies, we have also performed calcu- ily due to the larger KS orbital-energy differences and
lations with the KLI and KLICS potential, presented clearly visible in Fig. 1. There we have plotted the
in Tables 3 and 4, respectively. As before, the xc resulting excitation energies for the 2su ! 1pg trans-
kernels were approximated by the ALDA. Hence the ition in LDA and KLI approximation in order to
changes in the calculated excitation energies as visualize the effect of the various truncation approx-
compared to the results of Table 2 are solely due to imations on the resulting excitation energies. Finally,
the different KS potentials used. The columns headed the KLICS scheme leads to slightly larger orbital-
“full” display solutions of Eq. (17) taking into account energy differences and excitation energies than the
 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367 361

Table 5
Excitation energies for N2 from various methods at R ˆ 2:0744 a.u. The ALDA was employed for the xc kernels in the DFT calculations. Dev
denotes the mean absolute deviation from the experimental values for all 12 states. All numbers in mHartrees

State LDA a KLI b KLICS c LB d KLI 1 LB MRCC-SD e Expt f

2
A 3 Su 1 296.7 312.8 312.9 267.9 290.4 277.8 284.5
B 3Pg 280.1 298.7 297.6 262.4 280.6 296.0 295.5
W 3Du 327.6 343.2 343.8 305.8 324.5 328.1 326.3
a 1Pg 339.4 358.7 358.1 319.0 338.9 340.8 342.1
B 0 3 Su 2 355.8 370.9 371.9 337.4 354.2 362.5 355.4
a 0 1 Su 2 355.8 370.9 371.9 337.4 354.2 370.7 364.6
w 1Du 377.6 392.0 393.4 360.9 376.5 387.4 377.4
C 3Pu 383.7 406.4 404.7 369.7 388.1 411.2 411.2
E 3 Sg 1 379.8 439.7 456.1 452.7 446.2 431.9 441.0
a 00 1 Sg 1 385.3 458.1 476.3 – – 448.3 448.3
c 1Pu 435.3 480.3 500.8 – – 471.9 474.1
o 1Pu 509.8 526.2 525.9 – – 504.0 500.9
Dev 20.0 12.4 17.1 (23.1) (7.4) 4.0
a
Full matrix neglecting continuum states.
b
Full matrix using all occupied and the lowest 9 unoccupied orbitals.
c
Full matrix using all occupied and the lowest 9 unoccupied orbitals.
d
From Ref. [36].
e
From Ref. [43].
f
From Ref. [43].

KLI approximation, thus shifting the calculated is 43.0 mHartrees for the four higher states shown in
results further away from the experimental ones in Table 5 and Fig. 2. A similar picture is found for the
most cases. KLICS results. Here, the mean absolute errors are
The excitation energies calculated using the three 13.8 and 23.7 mHartrees for the eight lowest and
DFT approaches are compared in Table 5 and Fig. 2 four higher states, respectively. From the DFT
with results from a multi-reference coupled cluster methods, only the use of the KLI potential leads to
(MRCC) calculation from Ref. [43]. On an average, results with comparable accuracy for all states. In this
the KLI potential leads to the best results of the three approximation, the errors are 13.3 mHartrees for the
DFT methods with a mean absolute deviation of eight lowest and 10.7 mHartrees for the four higher
12.4 mHartrees from the experimental values. The excited states. While the excitation energies tend to be
use of the LDA potential gives a deviation of overestimated by the KLI and KLICS potentials, they
20.0 mHartrees whereas the KLICS results lie in tend to be underestimated by the LB potential. This is
between, showing a deviation of 17.1 mHartrees. especially true for the lowest eight states, where the
For comparison, we have also listed excitation ener- errors of both potentials are of opposite sign and are at
gies given in Ref. [36], which were obtained from the the same time almost equal in magnitude. This is illu-
model potential of van Leeuwen and Baerends [63] strated by column 6 of Table 5, where the arithmetic
(LB). The corresponding results deviate by mean of the KLI and the LB energies are displayed.
23.1 mHartrees from the experimental spectrum. The values given in this column are close to the
Clearly, the computationally much more expensive experimental spectrum (with a mean absolute
MRCC calculations lead to considerably better results deviation of 7.4 mHartrees). However, the KLI poten-
with a deviation of 4.0 mHartrees. Furthermore, the tial alone seems to give the best representation of the
quality of the LDA results depends very sensitively on higher Rydberg states. Of course, the MRCC results
the state under consideration: for the eight lowest show consistent accuracy for all states as well: For the
excited states, the mean absolute deviation from the eight lowest states, the error is 4.2 mHartrees, for
experimental values is only 8.6 mHartrees, whereas it the higher ones it is 3.6 mHartrees. Nevertheless, the
362 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367

Fig. 2. Difference of experimental and calculated excitation energies of N2 corresponding to Table 5. On the x-axis, the excited states are ordered
according to their experimental excitation energy with the energy increasing from left to right in arbitrary units.

Table 6
Orbital energies for CO at R ˆ 2:1322 a.u. from various DFT
quality of the DFT results is astounding, considering
approaches the simplicity of the approach. Especially, the relatively
small deviation of the eight lowest states obtained with
xcLDA KLI KLICS the LDA potential is impressive. We note that a third-
ETOT 2112.4782 2112.7832 2113.2616 order many-body perturbation theory calculation gives
Occupied orbitals: an error of the same magnitude for these states [43].
1s 218.7186 219.0959 219.1349 A similar picture is found for CO. The orbital ener-
2s 29.9072 210.2983 210.3406 gies resulting from calculations using the LDA, KLI
3s 21.0753 21.3280 21.3607
and KLICS potentials are shown in Table 6. As for N2,
4s 20.5216 20.7599 20.7920
1p 20.4455 20.6738 20.7067 the LDA potential decreases too rapidly in the valence
5s 20.3351 20.5526 20.5812 region, leading to only two bound states and a value
Unoccupied orbitals: for e HOMO of 2335 mHartrees. This is far above the
2p 20.0829 20.2853 20.3159 experimentally observed first ionization potential of
6s 20.0019 20.1631 20.1774
2515 mHartrees [64]. The KLI value of 2553 mHar-
3p .0 20.1208 20.1304
7s .0 20.1203 20.1292 trees is much closer, whereas adding the CS correla-
8s .0 20.0772 20.0845 tion potential moves the orbital energy further from
1d .0 20.0771 20.0836 the exact one, yielding 2581 mHartrees. Again, on
4p .0 20.0684 20.0721 average, the results from using the KLI potential are
9s .0 20.0639 20.0666
closest to the experimental values.
10s .0 20.0530 20.0550
In Table 7 the excitation energies resulting from an
 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367 363

Table 7 Table 9
Excitation energies for CO from an xcLDA-calculation at Excitation energies for CO from a KLICS-calculation at
R ˆ 2:1322 a.u. The LDA was employed for Vxc and the ALDA R ˆ 2:1322 a.u. The CS approximation for correlation added to
for the xc kernels. Dv KS denotes the KS orbital energy difference. the exact exchange was employed for Vxc in KLI-approximation
All numbers in mHartrees and the ALDA for the xc kernels. Dv KS denotes the KS orbital
energy difference. All numbers in mHartrees
State Dv KS SPA SMA Full a Exp b
State Dv KS SPA SMA Full a Exp b
A 1P 5s ! 2p 252.3 326.8 318.2 310.2 312.7
a 3P 223.8 222.0 221.4 232.3 A 1P 5s ! 2p 265.2 345.0 335.7 326.2 312.7
B 1S 1 5s ! 6s 333.2 338.9 338.6 338.0 396.2 a 3P 234.3 232.3 231.4 232.3
b 3S 1 331.5 331.5 331.6 382.2 I 1 S2 1p ! 2p 390.8 390.8 390.8 390.8 363.1
I 1 S2 1p ! 2p 362.6 362.6 362.6 362.6 363.1 e 3 S2 390.8 390.8 390.8 363.1
e 3 S2 362.6 362.6 362.6 363.1 a 0 3 S1 348.2 345.6 344.8 312.7
a 0 3 S1 318.1 315.0 314.9 312.7 D 1D 408.1 407.7 407.7 375.9
D 1D 381.2 380.7 380.7 375.9 d 3D 369.5 368.8 368.7 344.0
d 3D 340.4 339.6 339.6 344.0 B 1S 1 5s ! 6s 403.8 425.3 424.8 417.5 396.2
c 3P 4s ! 2p 438.8 420.4 420.0 420.2 424.5 b 3S 1 397.5 397.4 397.2 382.2
E 1P 1p ! 6s 443.6 443.5 443.5 443.5 423.7 E 1P 5s ! 3p 450.7 449.6 449.6 449.7 423.7
a
c 3P 447.9 447.9 447.5 424.5
b
Neglecting continuum states. C 1S 1 5s ! 7s 451.9 459.3 459.2 457.2 418.9
From Ref. [65]. j 3S 1 445.7 445.6 445.6 415.3
F 1S 1 5s ! 8s 496.7 499.8 499.8 497.0 455.7

LDA calculation are shown. The SPA and SMA lead a

Using all occupied and the lowest 9 unoccupied orbitals.
b
to results close to the ones from the full solution of the From Ref. [65].
matrix Eq. (17) taking into account all nine bound
orbitals. Generally, the SMA shows the smaller
energy. On the whole, it is apparent that excitation
error. Again, the largest deviations arise for states
energies involving transitions to the 2p KS orbital
such as A 1P where the corresponding KS orbital
are reproduced very well, whereas transitions to the
energy difference is far away from the exact excitation
6s orbital show a larger error. A glance at the orbital
energies given in Table 4 indicates that while the KS
Table 8
Excitation energies for CO from a KLI-calculation at
orbital energy difference is of comparable magnitude
R ˆ 2:1322 a.u. The exact exchange was employed for Vxc in for the transitions to the 2p KS orbital in all approx-
KLI-approximation and the ALDA for the xc kernels. Dv KS denotes imations, it is quite different for the transitions to the
the KS orbital energy difference. All numbers in mHartrees 6s orbital. For the latter, the LDA orbital-differences
State Dv KS SPA SMA Full a Exp b
are substantially smaller than the ones calculated from
the more accurate KLI potential, leading to excitation
A 1P 5s ! 2p 267.4 345.8 336.8 327.2 312.7 energies which are too low.
a 3P 237.0 235.0 234.2 232.3 Consequently, this should not be the case, if the
I 1 S2 1p ! 2p 388.6 388.6 388.6 388.6 363.1
KLI or KLICS potentials are used instead of the
e 3 S2 388.6 388.6 388.6 363.1
a 0 3 S1 346.7 344.2 343.4 312.7 LDA. This is clearly visible from Tables 8 and 9.
D 1D 405.6 405.2 405.2 375.9 Most notably, the energetic ordering of the orbital-
d 3D 367.6 367.0 366.9 344.0 energy differences corresponding to the 1p ! 2p
B 1S 1 5s ! 6s 389.6 409.4 408.9 402.4 396.2 and the 5s ! 6s transition is reversed compared to
b 3S 1 383.8 383.7 383.5 382.2
the LDA calculation. Furthermore, in KLI and KLICS
E 1P 5s ! 3p 431.8 430.8 430.8 431.1 423.7
c 3P 429.2 429.2 429.1 424.5 calculations the c 3P and E 1P excited states are
C 1S 1 5s ! 7s 432.3 437.7 437.7 436.4 418.9 assigned to correspond to a 5s ! 3p KS transition,
j 3S 1 427.0 427.0 427.0 415.3 whereas they arise from the 4s ! 2p and 1p ! 6s
F 1S 1 5s ! 8s 475.4 477.8 477.8 475.5 455.7 transitions, respectively, if the LDA potential is used.
a
Using all occupied and the lowest 9 unoccupied orbitals. Compared to the LDA, the quality of the results for
b
From Ref. [65]. these higher lying excited states is increased
364 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367

Table 10
Lower excitation energies for CO from various methods at R ˆ 2:1322 a.u. The ALDA was employed for the xc kernels in the DFT calculations.
Dev denotes the mean absolute deviation for all 11 states. All numbers in mHartrees

State LDA a KLI b KLICS c LB d MR-CCSD e SOPPAf Expt g

a 3P 221.4 234.2 231.4 205.1 232.3 219.4 232.3

A 1P 310.2 327.2 326.2 293.3 321.2 310.9 312.7
a 0 3 S1 314.9 343.4 344.8 307.2 308.3 293.6 312.7
d 3D 339.6 366.9 368.7 337.0 343.2 328.5 344.0
I 1 S2 362.6 388.6 390.8 – 371.5 356.5 363.1
e 3 S2 362.6 388.6 390.8 362.3 366.4 354.6 363.1
D 1D 380.7 405.2 407.7 – 377.0 366.0 375.9
dev 3.7 21.5 22.6 (12.0) 3.8 10.6
a
Full matrix neglecting continuum states.
b
Full matrix using all occupied and the lowest 9 unoccupied orbitals.
c
Full matrix using all occupied and the lowest 9 unoccupied orbitals.
d
From Ref. [36].
e
From Ref. [66].
f
From Ref. [66].
g
From Ref. [65].

significantly if the KLI potential is used. However, the If all 11 states are taken into account, the conven-
KLICS potential overestimates the resulting excita- tional quantum chemistry methods are superior on
tion energies considerably, especially for the higher average, with a mean absolute deviation well below
lying states. For both approaches, the SPA and SMA 10 mHartrees for the eleven states listed. The DFT
give results in close agreement with the ones from the approaches show a larger error, worst among them
full solution of Eq. (15), except for the 5s ! 2p sing- the KLICS results with 22.2 mHartrees. The LDA
let transition where the KS orbital difference is far and KLI potentials lead to an almost equivalent
from the experimental value. average deviation of 14.4 and 15.4 mHartrees,
In Table 10 and Fig. 3 we compare our results for respectively.
the seven lowest excited states of CO with those
obtained from the MRCC method and the second
order polarization propagator approach (SOPPA) 5. Summary and conclusion
[66]. Here, the LDA leads to the best results with an
average mean absolute deviation from the experimen- The main purpose of this work was to study the
tal values of 3.7 mHartrees, whereas the MRCC performance of various approximations involved in
approach leads to an error of 3.8 mHartrees. For the calculation of molecular excitation energies from
these lower states, both the KLI and KLICS again time-dependent DFT. Starting from the (unphysical)
overestimate the excitation energies with a deviation KS spectrum, we obtained corrections towards the
of 21.5 and 22.6 mHartrees, respectively. The LB- physical excitation energies for the N2 and the CO
potential underestimates the transition energies, but molecule.
with a mean deviation of only 12 mHartrees for the First of all, the calculation of response properties,
states shown. which in principle involves an infinite number KS
However, for the higher states a different picture is orbitals, requires a truncation of the problem in one
found, as may be seen from Table 11 and Fig. 3. For way or another. For the excitation energies studied in
these states, the KLI results are best with a mean this work, the single-pole approximation (SPA), which,
absolute deviation of 4.9 mHartrees, which is better in a nondegenerate situation, only requires one occupied
than the MRCC results, which do show an error of (initial) and one virtual (final) KS orbital, already gives
5.4 mHartrees. The LDA performs poorly leading to results which are quite close to more refined approxima-
an average error of 33.2 mHartrees. tions (“full”) using more configurations. Since the SPA
 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367 365

Fig. 3. Difference of experimental and calculated excitation energies of CO corresponding to Tables 10 and 11. On the x-axis, the excited states
are ordered according to their experimental excitation energy with the energy increasing from left to right in arbitrary units.

allows a simple assignment of excitation energies, it can correct asymptotic behaviour of the xc potential is
serve as a first orientation in practical calculations. essential. In this regime, orbital functionals based on
Next, we have calculated excitation energies using exact exchange in the approximation of Krieger, Li
different exchange-correlation potentials. In this and Iafrate (KLI) performed very well. However,
context, the LDA was tested against (self-interaction these potentials show a tendency to overestimate the
free) orbital approximations. Our calculations show, molecular excitation energies. The inclusion of
that in order to obtain spectra from DFT which are correlation contributions in the form of Colle and
close to experiment, the underlying KS eigenvalue Salvetti (CS) consistently worsened the results,
differences have to be well represented. indicating that the CS correlation potential needs
In agreement with the results of Casida et al. [36], it improvement for the calculation of molecular proper-
was found that the LDA potential yields excellent ties. This overestimation is most pronounced for
results for lower excitation energies of molecules. lower excitation energies, which in turn are very
For these excitations, the relatively large self-interac- well represented by the LDA.
tion errors, which are present in the LDA-orbital ener- On the whole, the quality of the results obtained
gies, cancel to a large extent. This finally leads to a with the DFT scheme for excitation energies is
fairly good representation of the true KS eigenvalue very encouraging. Improvements are however
differences. necessary for the correlation potential. In our
However, this cancellation of errors ceases to work opinion, orbital functionals offer a viable route
for excitations to higher lying states. There, the in this direction.
366 T. Grabo et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 353–367

Table 11 [15] L.N. Oliveira, E.K.U. Gross, W. Kohn, Phys. Rev. A 37

Higher excitation energies for CO from various methods at (1988) 2821.
R ˆ 2:1322 a.u. The ALDA was employed for the xc kernels in [16] A. Nagy, Phys. Rev. A 42 (1990) 4388.
the DFT calculations. Dev denotes the mean absolute deviation [17] A. Nagy, J. Phys. B 24 (1991) 4691.
for all 11 states. All numbers in mHartrees [18] A. Nagy, Phys. Rev. A 49 (1994) 3074.
[19] A. Nagy, Int. J. Quantum Chem. 56 (1995) 225.
State LDA a KLI b KLICS c MR-CCSD d SOPPAe Expt f [20] A. Nagy, Int. J. Quantum Chem. Symp. 29 (1995) 297.
[21] A. Nagy, J. Phys. B 29 (1996) 389.
b 3 S1 331.6 383.5 397.2 388.1 381.1 382.2 [22] A. Nagy, Adv. Quant. Chem. 29 (1997) 159.
B 1 S1 338.0 402.4 417.5 405.7 398.0 396.2 [23] I. Andrejkovics, A. Nagy, Chem. Phys. Lett. 296 (1998) 489.
E 1P 443.5 431.1 449.7 429.2 422.2 423.7 [24] M. Petersilka, Diplomarbeit, Universität Würzburg, 1993.
c 3P 420.2 429.1 447.5 425.2 418.6 424.5 [25] M. Petersilka, E.K.U. Gross, Satellite Symposium of the 8th
dev 33.2 4.9 21.3 5.4 2.6 International Congress of Quantum Chemistry, Cracow,
a
Full matrix neglecting continuum states. Poland, Book of Abstracts, 1994.
b
Full matrix using all occupied and the lowest 9 unoccupied [26] E.K.U. Gross, 209th American Chemical Symposium of the
orbitals. Eighth International Congress of Quantum Chemistry (1995)
c
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d
From Ref. [66]. 76 (1996) 1212.
e
From Ref. [66]. [28] M. Petersilka, E.K.U. Gross, Int. J. Quantum Chem. 60 (1996)
f
From Ref. [65]. 181.
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