Excited-state spin-contamination in time-dependent density-functional theory for molecules with...

This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institution

and sharing with colleagues.

Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third party

websites are prohibited.

In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further information

regarding Elsevier’s archiving and manuscript policies areencouraged to visit:

http://www.elsevier.com/copyright

http://www.elsevier.com/copyright

Author's personal copy

Excited-state spin-contamination in time-dependent density-functional theoryfor molecules with open-shell ground states

Andrei Ipatov, Felipe Cordova, Loïc Joubert Doriol, Mark E. Casida *

Département de Chimie Moléculaire (DCM, UMR CNRS/UJF 5616), Institut de Chimie Moléculaire de Grenoble (ICMG, FR-2607), Université Joseph Fourier (Grenoble I), 301 rue de laChimie, BP 53, F-38041 Grenoble Cedex 9, France

a r t i c l e i n f o

Article history:Received 21 April 2009Received in revised form 23 July 2009Accepted 23 July 2009Available online 3 August 2009

Keywords:Time-dependent density-functional theory(TDDFT)Excited statesOpen-shell moleculesSpin-contamination

a b s t r a c t

While most applications of the linear response formulation of time-dependent density-functional theory(TDDFT) have been to the calculation of the excited states of molecules with closed-shell ground states,Casida’s formulation of TDDFT opened the way to TDDFT calculations on molecules with open-shellground states by allowing for different-orbitals-for-different-spin, Although a number of publicationshave now appeared applying TDDFT to molecules with open-shell ground states and give surprisinglygood results for simple excitations, it is relatively easy to show that some excited states of open-shellmolecules will have unphysically large amounts of spin contamination. There is thus a clear need forcomputational tools which can separate physical from unphysical excited spin states in TDDFT. Weaddress this need by using analytic derivative techniques to develop formulae for the 1- and 2-electronreduced density difference matrices, in essential agreement with those obtained by Rowe in the equa-tion-of-motion superoperator approach to Green’s functions in nuclear physics. The corresponding for-mula for excited-state spin contamination appears to be generally good enough for assigning excited-

state spin symmetries, but does lead to a small overestimation of bS2D E

in the cases considered here. This

(apparently small) problem is eliminated when the Tamm–Dancoff approximation is used in TDDFT.� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The computational efficiency of time-dependent density-func-tional theory (TDDFT) has made it a state-of-the-art method forthe theoretical description of the electronic absorption spectra oflarge molecules with closed-shell ground states. In contrast, mole-cules with open-shell ground states have absorption spectra whichare more difficult to describe with any method. Nevertheless indi-cations are that the TDDFT formulation of Casida [1] gives an excel-lent description of simple excitations in radicals [2–5]. It waspointed out in an earlier paper by one of us that conventionalTDDFT will also generate excited states with unphysically largeamounts of spin contamination [6]. The purpose of the present pa-per is to develop a formula permitting to distinguish these unphys-

ical states from states with physically reasonable values of bS2D E

for

TDDFT calculations carried out in a different-orbitals-for-different-spins (DODS) formalism. To the best of our knowledge, and afterextensive appropriate inquiries, no such formula has beenpresented before for excited states obtained from linear responsecalculations either within TDDFT or within time-dependent

Hartree–Fock (TDHF). [There is, however, a singles configurationinteraction (CIS) formula [7] to which we will refer at the appropri-ate moment.]

It hardly seems necessary to review the basics of TDDFT in avolume, such as this one, devoted to more specific aspects of thesubject. Suffice it to say, that this article is restricted to Casida’s for-mulation of linear-response TDDFT for calculating electronic exci-tation energies [1], which cast TDDFT into the random phaseapproximation (RPA) form, already familiar to quantum chemistsfrom TDHF. (Unfortunately RPA means a number of different thingsin the literature, including time-dependent Hartree linear responsetheory using Kohn–Sham orbitals and orbital energies, so someauthors prefer to use the term ‘‘Casida’s equations” in order to min-imize confusion.) For more information about TDDFT, the reader isreferred to anyone of a number of reviews on the subject [1,8–27]and to other articles in the present volume.

Most work has been for excitations in molecules with closed-shell ground states. Nevertheless the spectroscopy of moleculeswith open-shell ground states is also very interesting and, in par-ticular, one of our objectives is to adapt DFT to describe the groundand excited states of low-spin and high-spin iron(II) hexacoordi-nate complexes. This involves a number of challenges. So far wehave addressed primarily the spin-pairing energy problem for therelative energies of ground states with different spin [28–32].

0166-1280/$ - see front matter � 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.theochem.2009.07.036

* Corresponding author. Tel.: +33 4 76 63 56 28; fax: +33 4 76 63 59 83.E-mail address: [email protected] (M.E. Casida).

Journal of Molecular Structure: THEOCHEM 914 (2009) 60–73

Contents lists available at ScienceDirect

Journal of Molecular Structure: THEOCHEM

journal homepage: www.elsevier .com/locate / theochem


However, another problem which needs to be addressed is thequestion of the applicability of TDDFT for the calculation ofexcitation energies in open-shell systems. The original formulationof Casida’s equations [1] foresaw applications to molecules withopen-shell ground states in that the formulation included bothsymmetry-broken (i.e., different-orbitals-for-different-spins) andfractional-occupation-number ground states. The first applicationsto calculate excitations of molecules with open-shell ground statescame four years later [2–5,33–40]. Particularly significant was thatTDDFT seemed to provide the best simple theory for describingsimple excitations in radicals [2–5].

One of us has pointed out that adiabatic TDDFT must necessar-ily fail for describing more complex excitations whose descriptionrequires the explicit description of polyelectronic excited states[6]. This is because adiabatic TDDFT only includes singlet and trip-let spin-coupling operators which is insufficient for a completedescription of the excited states of open-shell molecules. In partic-ular, adiabatic TDDFT can only describe one-electron excitationsand the general description of bS2 spin eigenfunctions for the ex-cited states of molecules with open-shell ground states also re-quires the inclusion of spin permutations of one-electronexcitations, which are not necessarily themselves one-electronexcitations. An important consequence is that, for molecules withopen-shell ground states, one can at best trust only those excitedstates which preserve the expectation value of bS2, and even thenit is possible that some peaks will be missing.

We are working on polarization propagator corrections whichare nonadiabatic nonTDDFT corrections to adiabatic TDDFT whichwill remedy this problem [6]. Other approaches to the same prob-lem include the approximate corrections given by Van Duijnenet al. in their study of the visible spectrum of [FeIII(PyPepS)2]�

[41] and noncollinear spin-flip TDDFT [42–47].In the meantime, we turn our attention in the present paper to

the problem of better characterizing the reliability of conventionalTDDFT excited-state calculations for open-shell molecules by dis-cussing the problem of how to calculate spin contamination forTDDFT excited states.

2. Theoretical method

In this section we are concerned with the problem of calculatingthe spin contamination for the ground state Kohn–Sham determinantand for the excited-state multideterminantal TDDFT wave functionsarising from broken-symmetry calculations. Our treatment of theground state problem is not new (our final formula is the classic for-mula from Hartree–Fock theory which was also used, for example, inthe classic study of spin-contamination in DFT by Baker et al. [48] andwhich is the ‘‘noninteracting” model in another classic paper on spin-contamination in DFT by Wang et al. [49]), but needs to be reviewedto set the stage for our treatment of the excited-state problem.

Our notation is such that the system contains n" spin up elec-trons and n# spin down electrons for a total electron count of,

n ¼ n" þ n#: ð2:1Þ

In our (and most other) calculations spin up is always the majorityspin,

n" P n#: ð2:2Þ

Throughout the present paper we will adhere to the molecular orbi-tal index convention,

abc . . . fgh|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}unoccupied

j ijklmn|fflfflffl{zfflfflffl}occupied

j opq . . . xyz|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}free

; ð2:3Þ

where ‘‘free” indicates that the orbital is free to be either occupiedor unoccupied.

2.1. Spin contamination and reduced density matrices

The usual treatment of spin for multielectron wave functions isnot based upon broken-symmetry calculations [i.e., different-orbi-tals-for-different-spin (DODS)], but rather upon same-orbitals-for-different-spin (SODS). There is thus just one underlying basis ofmolecular orbitals (MOs) which is the same for both spins. Let usadopt this basis for the moment and express the many-electronspin operators in terms of second-quantized creation and annihila-tion operators, for which we use the shorthand,

ryr ¼ ayrr;

rr ¼ arr:ð2:4Þ

The many-electron azimuthal spin operator may be expressed interms of the spin-up and spin-down number operators,

bSz ¼12

n" � n#ð Þ;

nr ¼X

r

ryrrr;ð2:5Þ

from which it is easily seen that every single determinant is aneigenfunction of bSz with eigenvalue ðn" � n#Þ=2. In contrast, thesquare of the many-electron total spin angular momentum operatoris [50],

bS2 ¼ bP";# þ n" þ bSzbSz � 1� �

;bP";# ¼Xr;s

ry#sy"s#r":

ð2:6Þ

The operator bP";# permutes spin up and spin down electrons and isknown as the spin-transposition operator. This means that onlyhighest weight spin eigenfunctions (i.e., those with S ¼ MS) maybe represented as single determinants. The other spin eigenfunc-tions involve linear combinations of determinants correspondingto different permutations of spin up and spin down electronsamong the occupied orbitals. Spin contamination is the extent towhich the calculated expectation value of bS2 differs from the ex-pected eigenvalue of the same operator. A large amount of spin con-tamination is an indication that there is a problem in the calculationand so provides an a priori criterion for discarding a potentiallymeaningless result.

In this article, we are not so much concerned with the SODSrepresentation but rather with broken-symmetry calculationswhich allow different-orbitals-for-different-spins (DODS). Accord-ing to the basic principles of the usual Hohenberg–Kohn–Shamformulation [51,52], we should seek those orbitals which minimizethe total energy. Since this leads quite naturally to DODS whenspin-density functionals are used, it is hardly surprising that DODScalculations are the status quo in DFT. Indeed Pople et al. havestrongly argued in favor of DODS when treating open-shell mole-cules in DFT [53]. Wittbrodt and Schlegel have given convincingexamples showing that DODS DFT is superior to both SODS andspin-projected DODS for calculating potential energy surfaces[54]. This is also the case for treating open-shell molecules in DFT.

Spin-unrestricted Kohn–Sham calculations may in fact beequivalent to a reasonably good single or multideterminantal SODScalculation, and spin contamination provides one criterion in orderto judge if this is indeed the case. Let us then seek explicit formulaefor evaluating the expectation value of bS2. A critical characteristicin DODS calculations is that there are two orthonormal sets ofmolecular orbitals, constituting two complementary basis sets –one for each spin. We will denote the spin-down basis set by anoverbar to distinguish it from the spin-up basis set. (This is actuallyoverkill since only the overbar basis set will be used for spin-downfunctions, but it is intended as an explicit reminder that the spin-down molecular orbitals often differ significantly from the spin-up

A. Ipatov et al. / Journal of Molecular Structure: THEOCHEM 914 (2009) 60–73 61


molecular orbitals.) The overlap matrix between the two sets,Dr;�s ¼ hrj�si is known as the spin similarity matrix. It allows us to ex-pand one basis set in terms of the other,

ry# ¼X

�p

�py# �pjrh i

s# ¼X

�q

sj�qh i�q#:ð2:7Þ

Since DODS wave functions typically have a well-defined value ofMS ¼ ðn" � n#Þ=2, the crucial quantities to calculate when evaluatingspin contamination are the expectation values of the spin-up num-ber operator and the expectation value of the spin transpositionoperator. The spin similarity matrix only enters into the calculationof the later. After transformation into the DODS basis set, the spintransposition operator becomes,bP";# ¼ X

r;s;�p;�q

�py#sy"�q#r"hsj�qih�pjri: ð2:8Þ

Its expectation value is just,

bP";#D E¼X

r;s;�p;�q

�py#sy"�q#r"

D Ehsj�qih�pjri ¼

Xr;s;�p;�q

Cr"�q# ;�p#s" sj�qh ih�pjri; ð2:9Þ

where the two-electron reduced density matrix (2-RDM), C hasbeen introduced. [With this definition, C is normalized to n(n � 1) for an n-electron system.] This makes it clear how to evalu-ate spin contamination once we have the 2-RDM.

2.2. Ground-state spin contamination

The need for a 2-RDM poses a certain conceptual problem inDFT, since it is usually assumed that a wave function is neededto calculate the 2-RDM. There are two ways around this problem.

One way is to use model exchange-correlation holes, and hencemodel DFT density matrices. This approach has been taken byWang, Becke, and Smith [49] who used it to develop theexpression,

bS2D E

¼ n" � n#2

� � n" � n#2

þ 1� �

þZ

ms<0msðrÞdr; ð2:10Þ

where,

msðrÞ ¼ q"ðrÞ � q#ðrÞ ð2:11Þ

is the spin magnetization density. Cohen et al. have suggested ageneralization of the Wang–Becke–Smith formula for use with gen-eralized gradient approximations [55]. Schmidt, Shenvi, and Tullyhave used the Wang–Becke–Smith formula as a constraint duringDFT calculations in order to gain better control on property calcula-tions sensitive to spin contamination [56].

The other, older, approach is the one used in most DFT pro-grams. In this approach bS2

D Eis evaluated directly from the

Kohn–Sham reference determinant. The 2-RDM is,

Cqs;pr ¼ nqnp dq;pds;r � dq;rds;p� �

: ð2:12Þ

Given this 2-RDM, the formula for the 2-RDM may be applied di-rectly to the calculation of ground-state spin-contamination,

bP";#D E¼X

s;r;�p;�q

Cr"�q# ;�p#s" hsj�qih�pjri ¼ �Xocc

i;�j

hij�jih�jjii

¼ �Xocc

i;�j

jhij�jij2: ð2:13Þ

Then

bS2D E

¼ n" � n#2

� � n" � n#2

þ 1� �

�Xocc

i;�j

j i �j�� 2; ð2:14Þ

a formula which can be traced back to Löwdin in the context of Har-tree–Fock theory [57]. Wang, Becke, and Smith showed that Eqs.(2.10) and (2.14) give similar values for bS2

D E.

Since the Wang–Becke–Smith formula is based upon a single-determinantal model of the exchange-correlation hole, it is not ex-pected to apply to TDDFT. This is why our formulae for excited-statespin contamination in TDDFT will be derived as analytic derivatives ofTDHF excitation energies.

2.3. Reduced difference density matrices

The problem of spin contamination is similar but more compli-cated for excited states. In this case, one works with reduced differ-ence density matrices (RDDMs),

Dcp;q ¼ cIp;q � c0

p;q;

DCpq;rs ¼ CIpq;rs � C0

pq;rs:ð2:15Þ

We will obtain the RDDMs needed for excited-state spin contami-nation in TDDFT by taking derivatives of the TDHF excitation energyexpression,

Dcp;q ¼oxohq;p

;

DCpq;rs ¼ox

o½rpjsq�=2:

ð2:16Þ

It is convenient for present purposes to use the equations-of-motion(EOM) superoperator formalism for Green’s functions [58,59]. TheTDHF excitation energy expression can then be expressed in a com-pact fashion as,

x ¼ U bO; bH; bOyh ih i�� /D E; ð2:17Þ

where U is the Hartree–Fock determinant,

bH ¼Xp;q

hp;qpyqþ 12

Xp;q;r;s

½prjqs�pyqysr; ð2:18Þ

is the electronic hamiltonian, and

bOy ¼Xia

ayiXia þX

ia

iyaYia; ð2:19Þ

is the transition operator. The coefficients, Xia and Yia, are obtainedby solving the TDHF equation,

A BB� A�

� ~X~Y

!¼ x

1 00 �1

� ~X~Y

!; ð2:20Þ

using the normalization,

~X�� 2 � ~Y

�� 2 ¼ 1: ð2:21Þ

Here,

Aia;jb ¼ U iya; bH; byjh ih i�� UD E¼ di;jfa;b � da;bfj;i þ ½jbjai� � ½jijab�;

Bia;jb ¼ U iya; bH; jybh ih i�� UD E¼ ½bijaj� � ½bjjai�;

ð2:22Þ

where the matrix elements of the Fock operator are,

fp;q ¼ hp;q þX

i

½pqjii� � ½pijiq�ð Þ: ð2:23Þ

[It is perhaps worth remarking that Eq. (2.20) has paired excitationsolutions,

62 A. Ipatov et al. / Journal of Molecular Structure: THEOCHEM 914 (2009) 60–73


~X~Y

!$ x;

j~Xj2 � j~Yj2 ¼ 1;

ð2:24Þ

and de-excitation solutions,

~X~Y

!$ x;

~X�� 2 � ~Y

�� 2 ¼ �1:

ð2:25Þ

For the excitation energy solutions,

x ¼ xXX þxXY þxYX þxYY ;

xXX ¼Xia;jb

X�iaAia;jbXjb;

xXY ¼Xia;jb

X�iaBia;jbYjb;

xYX ¼Xia;jb

Y�iaB�ia;jbXjb;

xYY ¼Xia;jb

Y�iaA�ia;jbYjb;

ð2:26Þ

which is just another form of Eq. (2.17).]Straightforward differentiation of Eq. (2.17) gives

Dcp;q ¼ U bO; qyp; bOyh ih i�� UD E;

DCrq;ps ¼ U bO; pysyqr; bOyh ih i�� UD E:

ð2:27Þ

Of course,

x ¼Xp;q

hp;qDcq;p þ12

Xp;q;r;s

½prjqs�DCrs;pq: ð2:28Þ

Note that the two RDDMs are properly normalized with respect toone another because,X

q

DCrq;pq ¼ ðn� 1ÞDcr;p; ð2:29Þ

follows from the

n ¼X

q

qyq ð2:30Þ

form of the number operator and the fact that bOy bO� �is number

conserving. This is essentially the solution to our problem. It re-mains only to make it more explicit by evaluating the commutators,

DcX;Xp;q ¼

Xia;jb

X�iaXjb U iya; qyp; byh ih i�� UD E

¼ �np�nq

Xi

X�iqXip � npnq

Xa

X�paXqa;

DcX;Yp;q ¼

Xia;jb

X�iaYjb U iya; qyp; jybh ih i�� UD E

¼ 0;

DcY;Xp;q ¼

Xia;jb

Y�iaXjb U ayi; qyp; byjh ih i�� UD E

¼ 0;

DcY;Yp;q ¼

Xia;jb

Y�iaYjb U ayi; qyp; jybh ih i�� UD E

¼ �np�nq

Xi

Y�ipYiq � npnq

Xa

Y�qaYpa;

ð2:31Þ

where,

Dcp;q ¼ DcX;Xp;q þ DcX;Y

p;q þ DcY;Xp;q þ DcY;Y

p;q ; ð2:32Þ

and, where,

�n ¼ 1� np ð2:33Þ

is the hole occupation number corresponding to the particle occu-pation number, np. Putting it all together gives the 1-RDDM,

Dcp;q ¼ �npnq

Xa

X�paXqa þX

a

Y�qaYpa

!

þ �np�nq

Xi

X�iqXip þX

i

Y�ipYiq

!: ð2:34Þ

[Notice the presence of a minus sign in Eq. (2.21) but of a plus signin Eq. (2.34).] Appendix A gives explicit expressions for,

DCXXrq;ps ¼

Xia;jb

X�iaXjb Uj iya; pysyqr; byjh ih i

jUD E

;

DCXYrq;ps ¼

Xia;jb

X�iaYjb Uj iya; pysyqr; jybh ih i

jUD E

;

DCYXrq;ps ¼

Xia;jb

Y�iaXjb Uj ayi; pysyqr; byjh ih i

jUD E

;

DCYYrq;ps ¼

Xia;jb

Y�iaYjb Uj ayi; pysyqr; jybh ih i

jUD E

:

ð2:35Þ

Of course,

Crq;ps ¼ CXXrq;ps þ CXY

rq;ps þ CYXrq;ps þ CYY

rq;ps: ð2:36Þ

In Ref. [60], Eq. (2.27) is attributed to Rowe and apparently goesback to the very beginning of the equation-of-motion superoperatorapproach to Green’s functions in nuclear physics. Notice how theform of Eq. (2.27) is inherited from the form of the TDHF expressionfor the excitation energy [Eq. (2.17)]. At the risk of being overly sim-plistic, let us try to sketch a simplified version of how Rowe’s for-mula works in the equation-of-motion superoperator approach toGreen’s functions. The transition operator, bOy, as an approximationto the exact solution,bOy ¼ jIih0j; ð2:37Þ

of the equation-of-motion,

bH; bOyh i¼ xbOy: ð2:38Þ

With this assumption about bOy, it is easily seen that Rowe’s formulagives,

0 bO; bA; bOyh ih i�� 0D E¼ I bA�� ID E

� 0 bA�� 0D E¼ D bAD E

; ð2:39Þ

which is just what we want. [Furthermore,

0 bOy; bA; bOh ih i�� 0D E¼ 0 bA�� 0D E

� I bA�� ID E; ð2:40Þ

for de-excitations.]Although our derivation of Rowe’s formula is rigorous as far as it

goes, it is important to realize that Rowe’s formula it is in fact anapproximation. Of course, we do not really have the exact solution[Eq. (2.37)], but at best,

Xia ¼ 0 iyh

a; bOy�� iD �� 0h i ¼ h0jiyajIi;

Yia ¼ 0 ay�

i; bOy�� iD �� 0h i ¼ h0jayijIi;ð2:41Þ

which are more closely related to transition moments than toexpectation values. (In fact, transition moments are routinely usedin TDDFT to calculate oscillator strengths from transition dipolemoments [1].) Indeed Lynch et al. [60] have criticized Rowe’s for-mula because it misses terms which should be present in a moregeneral equation-of-motion superoperator expression for excited-state expectation values.

One should ask how Rowe’s formula might be corrected. Thereappear to be two distinct approaches. One is to abandon linear re-sponse theory and go to quadratic response theory where general



matrix elements appear between excited states [cf., Eq. (15) of Ref.[61].] Of course, this solution is hardly expected to help if our goalis to describe excited-state spin-contamination within linear re-sponse theory. Another way, more in line with the analytic deriva-tives of linear response theory approach taken here, is to realizethat we have completely neglected molecular orbital relaxation,and hence have only obtained the unrelaxed RDDMs. The fully re-laxed RDDMs can be obtained by using analytic derivative tech-niques, as has been done by Furche and Ahlrichs [62] in the caseof TDDFT. A new element then enters into the theory in terms ofthe Z vector obtained by solving a second coupled equation ofthe form,

ðAþ BÞ~Z ¼ ~R; ð2:42Þ

where the reader is referred to Ref. [62] for an exact definition of ~R.The fully relaxed 1-RDDM then becomes,

Dcp;q ¼ � npnq

Xa

X�paXqa þX

a

Y�qaYpa

!

þ �np�nq

Xi

X�iqXip þX

i

Y�ipYiq

!þ np�nqZpq þ �nqnpZ�pq ð2:43Þ

with corresponding corrections in the 2-RDDM.It should thus be clear that Rowe’s formula should be taken as

approximate in the context of the present work. Nevertheless itwill be seen to be reasonably accurate, at least for the examplesconsidered here and has the merit of being relatively inexpensivefrom a computational point of view.

2.4. Excited-state spin contamination

Plugging the explicit expression for the 2-RDDM into theequation,

D bS2D E

I¼ bS2D E

I� bS2D E

0¼X

r;s;�p;�q

DCr"�q# ;�p#s" hsj�qih�pjri; ð2:44Þ

gives our final result for the difference in hS2i between the Ith ex-cited state and the ground state. This is,

D bS2D E

I¼X�i;�j;k;�a

XI��i�a#X

I�j�a# kj�i � �jjk

�þXi;j;�k;a

XI�ia"X

Ija"hjj�kih�kjii

�X�i;j;�a;�b

XI��i�a#X

I�i�b#hjj�bih�ajji �

Xi;�j;a;b

XI�ia"X

Iib"haj�jih�jjbi

� 2RX�i;j;�a;b

XI��i�a#X

Ijb"hjj�iih�ajbi þ 2R

Xi;�j;a;�b

XI�ia"Y

I�j�b#haj�jih�bjii

þ 2RX�i;j;�a;b

XI��i�a#Y

Ijb"hbj�iih�ajji þ

X�i;�j;k;�a

YI��i�a#Y

I�j�a#hkj�jih�ijki

þXi;j;�k;a

YI�ia"Y

Ija"hij�kih�kjji �

X�i;j;�a;�b

YI��i�a#Y

I�i�b#hjj�aih�bjji

�Xi;�j;a;b

YI�ia"Y

Iib"hbj�jih�jjai � 2R

X�i;j;�a;b

YI��i�a#Y

Ijb"hbj�aih�ijji; ð2:45Þ

where R means to take the real part of the following expression.Let us check that this formula is reasonable. A first check is that

setting the Y components to zero should give the previous result ofMaurice and Head-Gordon for the TDHF Tamm–Dancoff approxi-mation (also known as CIS) [7], and this is indeed the case.

Another way to check the result is to see what happens whenthe general formula (2.45) is applied to the case of a closed-shellmolecule with SODS. Then n" ¼ n# and

hrj�si ¼ dr;s: ð2:46Þ

Eq. (2.45) reduces to,

D bS2D E

I¼X

i;a

XI�i�a#

�� 2 þXi;a

XIia"

�� 2 � 2RX

i;a

XI��i�a#X

Iia"

þX

i;a

YI�i�a#

�� 2 þXi;a

YIia"

�� 2 � 2RX

i;a

YI�i�a#Y

Iia": ð2:47Þ

For the singlet-coupled solutions,

X�i�a# ¼ Xia";

Y�i�a# ¼ Yia";ð2:48Þ

and,

D bS2D E

I¼ 0: ð2:49Þ

In contrast, for the triplet-coupled solutions,

X�i�a# ¼ �Xia";

Y�i�a# ¼ �Yia";ð2:50Þ

and

D bS2D E

I¼ 4

Xi;a

XIia"

�� 2 þ 4X

i;a

YIia"

�� 2: ð2:51Þ

This last equation can be written in spin-orbital notation as,

D bS2D E

¼ 2P

ia Xiaj j2 þ 2P

ia Yiaj j2Pia Xiaj j2 �

Pia Yiaj j2

: ð2:52Þ

For an excitation,

D bS2D E

¼ D bS2D E

I¼ bS2D E

I� bS2D E

0P 2; ð2:53Þ

which can exceed the expected value of two because of the normal-ization condition [Eq. (2.21)] unless the Tamm–Dancoff approxima-tion has been made (i.e., the Yia are zero), in which case,

D bS2D E

I¼ 2: ð2:54Þ

exactly. [For a de-excitation,

D bS2D E

¼ �D bS2D E

I¼ bS2D E

0� bS2D E

16 �2; ð2:55Þ

which can be less than the expected value of minus two because ofthe normalization condition [Eq. (2.25)]. This is a clear example ofwhy X and Y components must have the same sign in Rowe’s for-mula. There is an implicit denominator which changes sign on goingfrom an excitation to a de-excitation, which is exactly the way itshould be.]

The somewhat surprising observation that D bS2D E

may exceedtwo for triplet excitations in closed-shell molecules comes fromthe approximate nature of Rowe’s formula. However, as we willsee in the section after next, the differences between excited-statespin contamination for full TDDFT and for TDA TDDFT calculationsappears to be relatively small, and is certainly small enough thatour formulae have definite practical value.

3. Computational details

Calculations were carried out with two different computer pro-grams, namely GAUSSIAN [63] and a modified version of DEMON2K [64].Both programs carry out DFT calculations in a similar way and ingeneral give very similar results when the same calculation is per-formed with the two programs. This allows us to use the two pro-grams in a complementary way. For example, GAUSSIAN

automatically assigns the symmetries of the molecular orbitals.The particular version of DEMON2K used in the present work doesnot have this feature so we rely on comparison with the outputof Gaussian calculations for this aspect of our analysis. The partic-



ular version of GAUSSIAN used in the present work does not allowTDDFT calculations to be performed using the Tamm–Dancoffapproximation (TDA) while DEMON2K does have this feature. The cal-culation of spin-contamination for TDDFT excited states has beenimplemented using the unrelaxed density matrix in the programDEMON2K [64].

The implementation of TDDFT in GAUSSIAN has been described inRef. [65]. Our implementation of TDDFT in DEMON2K has been previ-ously described elsewhere [66]. It was found in that study that DE-

MON2K calculations using the GENA3* auxiliary basis set and FINE gridgave excitation energies within 0.02 eV of those given with Gauss-ian without the use of density fitting functions.

Calculations were carried out on five small molecules previ-ously used as a test set for TDDFT calculations by Hirata andHead-Gordon [2] and Guan et al. [33,35], namely BeH, BeF, CN,CO+, and Nþ2 , plus a sixth molecule, CH2O+, included in the testset of Guan et al. [35]. All of these molecules are radicals with adoublet ground state. The molecular geometries used were thesame as those experimental geometries previously used by Guanet al. [35]. The orbital basis set used was that of Sadlej [67–70].Extensive use was made of defaults in carrying out calculationswith Gaussian. Our DEMON2K calculations used the GenA3* auxiliarybasis set and the FINE grid.

Since both the present calculations and the previous calcula-tions of Guan et al. [33,35] used programs from the deMon suiteof programs, it may be useful to point out that the present version(DEMON2K) differs from the previous implementation used by Guanet al. in several ways, including different convergence criteria,grids, auxiliary basis sets, and the inclusion of a charge conserva-tion constraint in the auxiliary basis charge density fitting algo-rithm. Fig. 1 gives a visual idea of how excitation energies fromthe three programs compare on average. There is improved agree-ment between DEMON2K and GAUSSIAN than with respect to the previ-ous DEMON-DYNARHO [35] and GAUSSIAN. On the other hand, theagreement between DEMON2K and GAUSSIAN is not nearly as good asthat previously found for closed-shell molecules [66]. Little isknown about the source of these differences except that (i) it doesnot come from the orbital energies since these agree to better than0.05 eV and that (ii) differences between DEMON2K and GAUSSIAN

TDLDA excitation energies tend to correlate with differences be-tween DEMON2K TDLDA excitation energies with and without theTDA (Fig. 2.) Whatever their origin these differences between the

results of our GAUSSIAN and DEMON2K calculations are nevertheless suf-ficiently small so as to permit us to go on with our study of excited-state spin contamination and will not be further investigated here.

4. Results

Our goal in this section is to show how the calculation of spincontamination in spin unrestricted TDDFT calculations can helpwith assignments. The calculated order of spectroscopically activeLDA orbitals is shown in Table 1 for the five small molecules stud-ied here.

Before examining our TDLDA results in detail, let us first con-sider in a very general manner what type of excitations are possi-ble. In discussing our results, we will follow a common conventionwhen discussing radicals that the HOMO is the highest doublyoccupied molecular orbital, the SOMO is the singly occupiedmolecular orbital, and the LUMO is the lowest unoccupied molec-ular orbital. With this scheme in mind, excitations may be classi-fied according to whether they involve the SOMO (Fig. 3) orwhether the SOMO is more of a spectator to the excitation process(Fig. 4). The simplest excitations are those involving the SOMO.They are clearly doublets and are typically found among the lowestexcitations in radicals. Experimental data for the excitations de-noted as type (2) in Fig. 3 is readily available for cations sincethe excited state configuration is easily obtained as the principleionization out of a doubly-occupied orbital of the parent neutralmolecule. No particular problem with spin-contamination is antic-ipated with excitations involving the SOMO and the transitionsmay have nonzero oscillator strength.

Fig. 1. Correlation plot for excitation energies in eV calculated with GAUSSIAN andwith DEMON2K (closed symbols) and with DEMON-DYNARHO (open symbols) [35].

Fig. 2. Correlation plot for the difference between the TDA TDLDA and full TDLDAexcitation energies calculated with DEMON2K (y-axis) and the difference between theTDLDA excitation energies calculated with DEMON2K and with GAUSSIAN 03 (x-axis). Allenergies are in eV.

Table 1Ordering and occupation of those LDA occupied and unoccupied ground state orbitalswhich are most likely to participate in low-lying excitations. The singly occupiedmolecular orbital (SOMO) has been emphasized by putting it is bold face.

Molecule Configuration

BeH . . . 2r23r11p04r05r0 . . .

BeF, CN, CO+. . . 4r21p45r12p0 . . .

Nþ2 . . . 2r2u1p4

u3r1g1p0

g . . .

CH2O+. . . 1b2

15a212b1

22b015a0

13b02 . . .



The situation is more complicated for excitations where theSOMO is nominally a spectator (Fig. 4). We now have the typicalproblem of three coupled spins. As summarized in Table 2, thisgives rise to one quartet and two doublet states. Since ordinaryTDDFT calculations cannot flip spins only singlet and triplet cou-pled excitations are allowed,

1=ffiffiffi2p� �

�isaj � jis�a��

: ð4:1Þ

When the SOMO is a spectator to the excitation process and the sin-glet-coupled excitation has zero spin, the result is the MS ¼ þ1=2function of doublet 1. Thus the singlet-coupled excitation in TDDFThas a clear physical interpretation as a true eigenfunction of thespin operators. Moreover it may have nonzero oscillator strength.However, the case of the triplet-coupled excitation in TDDFT is lessstraightforward. This triplet-coupled excitation carries a spin of 1and so can couple with the SOMO spin of 1/2 to yield a quartetand a doublet. These are the quartet and second doubletMS ¼ þ1=2 functions shown in the table. They have zero oscillatorstrength. However, doublet 2 can mix with doublet 1 in order toproduce two excited doublet states with nonzero oscillatorstrength. Unfortunately this can never happen in conventionalTDDFT using the adiabatic approximation. The problem is that theactual coupling involves the spin-flip configuration (5) of Fig. 4. Thisconfiguration is nominally a two-electron excitation and as such isnot accessible with the adiabatic approximation of TDDFT. In the

absence of access to this configuration only the unphysical tripletexcited state of Eq. (4.1) will be found and must either be discardedor the effects of the missing configuration must be included pertur-batively via the introduction of integrals which do not arise natu-rally in TDDFT. This latter procedure has been used to advantageby Van Duijnen et al. in their study of the visible spectrum of[FeIII(PyPepS)2]� [41]. The ability to calculate bS2

D Efor excited states

is an important help for distinguishing singlet- and triplet-coupledexcited states, thus permitting intelligent decisions about how totreat them. It also permits the identification of TDDFT excited stateswhich are neither singlet- nor triplet-coupled, but rather so spincontaminated as to be total nonsense.

The ground state spin contamination of the molecules studiedin this paper is given in Table 3. In the absence of spin contamina-tion, SðSþ 1Þ ¼ 0:75. The largest spin contamination is found forthe carbon monoxide cation, with ð bS2

D E¼ 0:7620Þ. If we assume

that the quartet state ðSðSþ 1Þ ¼ 3:75Þ constitutes the primarysource of contamination in the carbon monoxide cation,

jCOþi ¼ CDjdoubleti þ CQ jquarteti; ð4:2Þ

then

0:7620 ¼ 0:75� jCDj2 þ 3:75jCQ j2; ð4:3Þ

and the Kohn–Sham wave function is 99.6% doublet in characterðjCDj2 ¼ 0:996Þ. This is an acceptably low level of spin contamina-tion for most purposes and certainly allows the state to be classifiedas primarily doublet in character.

The first several TDLDA excitation energies and spin contamina-tion, D bS2

D E(parentheses), for our five test molecules are given in

Tables 4–9. Also given are oscillator strengths [square brackets]and results from our TDA calculations. Assignments have been

Fig. 3. Excitations involving the SOMO.

Fig. 4. Excitation where the SOMO is a spectator.

Table 2Coupling of three spins in three different orbitals, i, s, and a.

S MS Wave function

Quartet3/2 +3/2 jisaj3/2 +1/2 1=

ffiffiffi3p� �ðj�ısaj þ jis�aj þ ji�sajÞ

3/2 �1/2 1=ffiffiffi3p� �ðji�s�aj � j�ı�saj þ j�ıs�ajÞ

3/2 �3/2 j�ı�s�aj

Doublet 11/2 +1/2 1=

ffiffiffi2p� �ðj�ısaj � jis�ajÞ

1/2 �1/2 1=ffiffiffi2p� �ðji�s�aj � j�i�sajÞ

Doublet 21/2 +1/2 1=

ffiffiffi6p� �

ðj�ısaj þ jis�aj � 2ji�sajÞ1/2 �1/2 1=

ffiffiffi6p� �ðji�s�aj þ j�i�saj � 2j�is�ajÞ

Table 3Ground state spin contamination in our DEMON2K LDA calculations. The spin contam-ination obtained from GAUSSIAN03 is identical to the number of significant figuresshown.

Molecule bS2D E

Multiplicity (2S + 1)a Spin contaminationb (%)

BeH 0.7503 2.0003 0.010BeF 0.7513 2.0013 0.043CN 0.7546 2.0046 0.15CO+ 0.7620 2.0120 0.40Nþ2 0.7514 2.0014 0.047CH2O+ 0.7512 2.0012 0.040

a Calculated assuming that bS2D E

= S(S+1).b Coefficient of next highest allowed spin component squared times 100%, cal-

culated assuming that all spin contamination comes from the next highest allowedspin component.



made based upon analysis of the output of the DEMON2K TDA calcu-lations using the orbital symmetry assignments from GAUSSIAN03.Some reference values are given in the last column. In this casewe have preferred reference values obtained from theoreticalcalculations since these often contain more detailed informationabout assignments with which we may compare our ownassignments.

Inspection of the tables shows that there are mainly three

values of D bS2D E

found for the listed excited states of these mole-

cules. These are D bS2D E

� 0, indicating singlet-coupling and hence

excitation to a doublet excited state, D bS2D E

� 2, indicating trip-

let-coupling (TC) and hence excitation to a well-defined but

unphysical state, and D bS2D E

� 1 which corresponds to states

which are too spin contaminated to be considered any further.Not shown here are higher excitations which show values of

D bS2D E

significantly further from noninteger values. Figs. 5 and 6

compare the values of D bS2D E

for full and TDA TDLDA calculations.

The comparison is of particular interest for the triplet excitation

energies since we have seen that the value of D bS2D E

may exceed

2 for full TDLDA calculations. Indeed triplet D bS2D E

values in the full

TDLDA calculations are always larger than the corresponding val-

ues from TDA TDLDA calculations. While the value of D bS2D E

some-

times exceeds 2 for full TDLDA calculations, this never seems to bethe case for TDA TDLDA calculations. On the whole, however, thisseems to be at most a minor problem and in no way undermines

Table 4Spin contamination in BeH excited states.

State Excitation energy (eV) [f] D bS2D E� �

TDLDAa TDLDAb TDLDA TDAc Assignmentc Reference valuesd

TDLDA spin down ionization threshold = 8.0 eV7 5.6414 5.4803 5.5030 3Rþ

[0.1684] (0.0034) (0.0039) ð3r! 6r;2r! 3rÞ6 5.6338 5.1685 5.2953 TCP

[0.0010] (2.0115) (1.9784) ð2r! 1p;2�r! 1�pÞ5 5.1318 4.8049 4.8676 2Rþ

[0.0029] (0.0297) (0.0303) ð3�r! 3�rSOMOÞ4 4.8451 4.7047 4.7104 2P 6.31

[0.0022] (0.0134) (0.0091) ð3rSOMO ! 5rÞ3 4.7576 4.6300 4.6481 2Rþ

[0.0022] (0.0134) (0.0091) ð3rSOMO ! 5rÞ

TDLDA spin up ionization threshold = 4.6 eV2 4.6633 4.5103 4.5278 2Rþ 5.51

[0.0491] (0.0083) (0.0063) ð3rSOMO ! 4rÞ1 2.3657 2.2479 2.2860 2P 2.56

[0.0427] (0.0060) (0.0011) ð3rSOMO ! 1pÞ

aGAUSSIAN03.

bDEMON2K.

cTDLDA TDA.

d From Table I of Ref. [66].

Table 5Spin contamination in BeF excited states.



TDLDA spin down ionization threshold = 10.41 eV7 6.7268 6.6988 6.7092 2P

[0.0027] (0.0155) (0.0114) ð5rSOMO ! 4pÞ6 6.5432 6.4884 6.4940 2D

[0.0000] (0.0011) (0.0009) ð5rSOMO ! 1dÞ5 6.1070 6.0584 6.0637 2Rþ

[0.0739] (0.0031) (0.0024) ð5rSOMO ! 8rÞ4 5.6201 5.5811 5.5917 2P

[0.0000] (0.0003) (0.0003) ð5rSOMO ! 7rÞ3 5.6536 5.5639 5.5666 2P

[0.0000] (0.0003) (0.0003) ð5rSOMO ! 3pÞ

TDLDA spin up ionization threshold = 5.4 eV2 5.3462 5.5264 5.3314 2Rþ 6.345

[0.0977] (0.0017) (0.0014) ð5rSOMO ! 6rÞ1 4.0747 3.9816 4.0225 2P 4.2049

[0.1693] (0.0000) (0.0005) ð5rSOMO ! 2pÞð1�p! 5�rSOMOÞ

aGAUSSIAN03.

bDEMON2K.

cTDLDA TDA.

d Extended configuration interaction singles (XCIS)results from Ref. [71].



the utility of the D bS2D E

calculation. (It also suggests that we could

just as well use the TDA formula to calculate D bS2D E

in full TDDFT

calculations after first appropriately renormalizing the X compo-nents of the TDDFT vector, though we have not done this here.)

The utility of calculating D bS2D E

is immediately evident in thecase of CN (Table 6). It is easily seen that only three out of the firstfive states have physical meaning. The remaining four states are TCstates which were misassigned in Ref. [35], for lack of the analytictools presented in the present work. (Other examples of misassign-ments can be found in the same article for CO+ and Nþ2 .) Note thatstate 3 could not have been identified as a TC state on the basis ofoscillator strength alone since it has nonzero oscillator strength.Once the TC states are eliminated, the remaining four states arein good agreement with reference values taken from the literature.

A second criterion for eliminating TDDFT excitation energies isthe position of the adiabatic TDDFT ionization continuum which

occurs at minus the orbital energy of the highest orbital of occu-pied by an electron of a given spin [77]. Were the exchange-corre-lation functional exact this would be the exact ionization potential.However, the LDA exchange-correlation potential goes too quicklyto zero at large distances, leading to artifictual underbinding of theelectrons and hence to a TDLDA ionization threshold which may betoo low by several eV. A more accurate estimate of ionization ener-gies may be obtained by DSCF calculations (not given here). Themajority spin is up in all of our calculations. A priori this suggeststhat the spin-up ionization potential should be smaller than thespin-down ionization potential. This is true for BeH, BeF, andCH2O+ where the spin up ionization potential is 3–5 eV lower thanthe spin down ionization potential. In CN, CO+, and Nþ2 , the spin upand spin down ionization potentials are nearly identical becausethey refer to degenerate p orbitals. It may or may not be significantthat the spin down ionization potential actually exceeds the spinup ionization threshold for CO+ by 0.4 eV. This is not the case for

Table 6Spin contamination in CN excited states.



TDLDA spin down ionization threshold = 9.5 eVTDLDA spin up ionization threshold = 9.7 eV7 8.3190 8.3210 8.3210

[0.0000] (0.9967) (0.9967)6 8.0324 8.0334 8.0334

[0.0000] (0.9967) (0.9967)5 8.0518 7.9829 8.0236 2P 8.619

[0.0027] (0.0948) (0.0964) ð5rSOMO ! 2pÞ4 7.4325 7.1807 7.2699 TCD

[0.0000] (2.0050) (1.9838) ð1p! 2p;1�p! 2�pÞ

3 6.6347 6.0967 6.4792 TCR+

[0.0008] (1.9928) (1.9152) ð1p! 2pÞ2 3.2333 2.8364 3.1908 2Rþ 3.661

[0.0029] (0.1566) (0.0786) ð4�r! 5�rSOMOÞ1 1.3576 1.2131 1.3017 2P 1.235

[0.0029] (0.0160) (0.0084) ð1�p! 5�rSOMOÞ

aGAUSSIAN03.

bDEMON2K.

cTDLDA TDA.

d Multireference configuration interaction vertical excitation energies calculated from the data in Table III of Ref. [72] at Re=2.211 bohr.

Table 7Spin contamination in CO+ excited states.




[0.0000] (0.9885) (0.9885)6 9.4114 9.4107 9.4107

[0.0000] (0.9884) (0.9884)5 8.9870 8.9203 8.9782 2P

[0.0118] (0.0950) (0.1102) ð5rSOMO ! 2pÞ4 8.8850 8.6319 8.7202 TCD

[0.0000] (1.9189) (1.8950) ð1p! 2p;1�p! 2�pÞ

2 5.0089 4.6249 4.8274 2Rþ 5.965[0.0176] (0.1941) (1.8785) ð4�r! 5�rSOMOÞ

1 3.1623 2.9126 3.0144 2P 3.137[0.0038] (0.0049) (0.0074) ð1�p! 5�rSOMOÞ

aGAUSSIAN03.

bDEMON2K.

cTDLDA TDA.

d Full configuration interaction with the Huzinaga–Dunning [4s2p] basis set, calculated from the data in Table 2 of Ref. [73].



CN (where the spin up ionization potential exceeds the spin-downionization threshold by 0.2 eV) nor for Nþ2 (where the spin-up ion-ization threshold by 0.2 eV). While ground-state v-representability

does not always hold for open-shell molecules (this point is dis-cussed in detail in Ref. [78]), we tend to believe that the major dif-ference between these molecules in our calculations is associated

Table 8Spin contamination in Nþ2 excited states.




[0.0000] (0.9999) (0.9999)6 9.2685 9.2617 9.2617

[0.0000] (0.9999) (0.9999)5 9.2685 9.1901 9.2415 2Pg

[0.0000] (0.0016) (0.0016) ð3rSOMOg ! 1pgÞ

4 8.3921 8.1080 8.2238 TCDu

[0.0000] (2.0207) (1.9965) ð1pu ! 1pg ;

1�pu ! 1�pgÞ3 7.3319 6.7385 7.2329 TCRþu

[0.0359] (0.1954) (0.0985) ð2�ru ! 3�rSOMOg ;

1�pu ! 1�pg ;

1�pu ! 1�pgÞ2 3.7195 3.2982 3.7822 2Rþu 3.57

[0.0359] (0.1954) (0.0985) ð2�ru ! 3�rSOMOg Þ

1 1.4634 1.3345 1.4321 2Pu 1.41[0.0021] (0.0191) (0.0081) ð1�pu ! 3�rSOMO

g Þ

aGAUSSIAN03.

bDEMON2K.

cTDLDA TDA.

d Configuration interaction from Ref. [74].

Table 9Spin contamination in CH2O+ excited states.


TDLDAa TDLDAb TDLDA TDAc Assignmentc Reference values

TDLDA spin up ionization threshold = 15.1 eVTDLDA spin down ionization threshold = 18.2 eV10 9.9685 9.9403 10.0286 2B2

[0.0005] (0.0127) (0.0067) ð2bSOMO2 ! 3b2Þ

9 9.7157 9.7036 9.7862 2B2

[0.0396] (0.0380) (0.0367) ð2bSOMO2 ! 3b2;

5a1 ! 6a1Þ8 8.6708 8.3665 8.5006 TCB2

[0.0000] (2.0046) (1.9834) ð5a1 ! 6a1;5�a1 ! 6�a1Þ7 7.9952 7.8765 7.8980 2B1 3.86d, 3.84e

[0.0006] (0.0058) (0.0047) ð1�b1 ! 2�bSOMO2 Þ

6 7.9510 7.8765 7.8980 TCB2

[0.0000] (1.9870) (1.9619) ð5a1 ! 6a1;

5�a1 ! 6�a1Þ5 7.5641 7.5623 7.7839 2A2

[0.0391] (0.0061) (0.0050) ð5a1 ! 2b1;

5�a1 ! 2�b1Þ4 5.4543 5.1002 5.2638 TCA2

[0.0001] (0.0064) (1.9933) ð5a1 ! 2b1;

5�a1 ! 2�b1Þ3 5.1707 5.0833 5.1678 2A1

[0.0145] (2.0502) (0.0045) ð2bSOMO2 ! 6a1Þ

2 4.7445 4.6046 4.6862 2A1 5.30d, 5.46e

[0.0083] (0.0019) (0.0019) ð5�a1 ! 2�bSOMO2 Þ

1 2.7014 2.6439 2.6669 2B1 5.78d, 6.46e

[0.0000] (0.0020) (0.0012) ð2bSOMO2 ! 2b1Þ

aGAUSSIAN03.

bDEMON2K.

cTDLDA TDA.

d Multireference configuration interaction calculation from Ref. [75].e Multireference configuration interaction calculation from Ref. [76].



with the larger amount of ground-state spin contamination in CO+

as opposed to CN and Nþ2 (Table 3). For the states considered here,the energy of the TDDFT ionization continuum is only a problemfor BeH (Table 4) and BeF (Table 5) and then only for excitationsinvolving spin-up electrons. This accounts for the very large errorin the 2Pð3rSOMO ! 5rÞ excitation energy of BeH with respect tothe reference value from the literature.

All things taken into consideration the remaining physicallymeaningful excitation energies are seen to be almost exclusivelysimple excitations involving the SOMO of the type illustrated inFig. 3. Without doubt this is one of the main reasons that TDDFThas been found to work well for the description of excitation ener-gies in radicals. An exception may be state 5 of CH2O+ (Table 9), forwhich we were unable to find any comparison data, but which theTDLDA predicts to have significant oscillator strength.

The case of CH2O+ should also serve as a final word of warning.Thanks to the details of the ab initio calculations given in the refer-ence literature, we are able to give a detailed correspondence be-tween our excitation energies and the ab initio excitationenergies, assuming that the orbitals have the same meaning inthe two types of calculation. However, we know of no theoremwhich says that this should be so. Had we simply relied on the totalsymmetry we would have exchanged the reference values for thefirst and seventh states and found much better agreement betweenTDDFT and the ab initio theory. Even comparison of the TDDFT andab initio potential energy curves given in Ref. [35] is not able to clar-ify this point.

It may be worth mentioning a different, pragmatic, approach toassigning excited-state spin-symmetry. In DEMON2K, the basic TDDFTEq. (2.20) is solved in the form [1],

X~F ¼ x2~F; ð4:4Þ

with,

X ¼ A� Bð Þþ1=2 Aþ Bð Þ A� Bð Þþ1=2: ð4:5Þ

In principle,

~F ¼ A� Bð Þ�1=2 ~X þ~Y� �

; ð4:6Þ

but is renormalized so that,

~Fy~F ¼ 1: ð4:7Þ

We have observed that the X matrix preserves all the symmetries

of the A matrix in the TDA. This suggests calculating D bS2D E

using

the TDA formula of Maurice and Head-Gordon, but ~F in place of ~X.

We call this, D bS2D E

F, though we have no strong theoretical reason

to expect this to give us a reasonable estimate of D bS2D E

. Let us be

quite clear about this point: There are two types of spin contamina-tion. The first type results from the absence of doubly and higherexcitations needed to correctly describe spin-permuted configura-tions. This type of spin-contamination may be called ‘‘trivial” in

so far as possible D bS2D E

values can be deduced in advance within

a SODS model. Any spin contamination beyond this is ‘‘nontrivial.”As the results shown in this paper indicate, the most serious spincontamination routinely encountered in TDDFT is of the trivial typeand its identification allows us to assign spin symmetries and re-

move or improve nonphysical solutions. Since the structure of~F re-

flects system symmetries, D bS2F

D Ewill give the expected value of

D bS2D E

in cases of trivial spin-contamination. However, D bS2F

D Emay not be a reliable estimate of nontrivial spin contamination,

though preliminary numerical tests suggest that the TDLDA D bS2F

D Eis numerically very similar to the corresponding TDA value. Wecan rationalize this observation using the ansatz proposed by Casidafor assigning TDDFT solutions [1]. According to this ansatz, the CISpart of the excited-state wave function is given by,

WI ¼Xiar

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ar � �ir

xI

rFI

iarayrirUþ � � � : ð4:8Þ

which allows us to identify ~F with the CIS vector ~X provided

�ar � �ir � xI for the dominant components. The use of D bS2D E

Fis

illustrated in Table 10. D bS2D E

Fis exactly equal to two for triplet

excitations from a closed-shell ground state. This is illustrated forstretched H2 and twisted C2H4 which show triplet instabilitiesðx2 < 0Þ associated with biradicaloid character. In these cases the

Fig. 5. D bS2D E

correlation graph for full and TDA TDLDA DEMON2K singlet excitationenergies.

Fig. 6. D bS2D E

correlation graph for full and TDA TDLDA DEMON2K triplet excitationenergies.



lowest TDDFT excited state (x2 is lowest) is a triplet excited state

with x imaginary. In this case D bS2D E

Fis exactly two, as it should

be for a triplet excitation. However, the TDDDFT D bS2D E

> 2, indicat-

ing (quite correctly) that there is a problem (i.e., symmetry breakingof the ground state orbitals will lead to a lower energy solution.) Ta-ble 10 also contains some selected nonphysical solutions of the

TDDFT problem where trivial spin contamination gives DhbS2i.(Numbers differ slightly from previous tables due to evolutions in

the numerics of DEMON2K.) The D bS2D E

Fresult is found to be system-

atically closer to the corresponding TDA quantity.In practice, we recommend outputting two values of D bS2

D Efor

full TDDFT calculations—the value of D bS2D E

obtained from Eq.

(2.45) with ~Y–0 because we have been able to derive it from first

principles within TDHF, and the value of D bS2D E

Fbecause it reliably

reflects the symmetry properties of the ~F vector.

5. Conclusion

Linear response TDDFT calculations of the electronic excitationsof molecules with open-shell ground states began within aboutfive years after the first implementations of Casida’s equations inquantum chemistry programs [2,4,33–36]. In many cases the re-sults were very encouraging. However, caveat emptor.1

As pointed out by Casida in his seminal paper reformulating lin-ear response TDDFT for the calculation of molecular electric excita-tion spectra [1], the TDDFT adiabatic approximation limits TDDFTto the description of single-electron excitations (albeit single-elec-tron excitations dressed to include correlation effects). As shownhere and elsewhere [6,79], this means that TDDFT calculations ofthe electronic excitation spectra of molecules with open-shellground states are frought with dangers. In particular the triplet-coupled excited states of radicals are clearly unphysical and somust either be discarded or corrected to make them physical andthere may be too few states of doublet symmetry. This work pre-sents equations for calculating the spin contamination of TDDFTexcited states, thus presenting an analytic tool which can helpthe user to better interpret the results of his or her TDDFT calcula-tions for open-shell molecules.

Our formulae for the calculation of spin contamination havebeen used to analyze low-lying excitations in five small radicals.The occurrence of unphysical triplet-coupled states has beenclearly demonstrated. Once these were eliminated from furtherconsideration the remaining excitations were found to be of a par-

ticularly simple type involving either excitation into the singlyoccupied molecular orbital (SOMO) or excitation out of the SOMO,and the associated TDDFT excitation energies are completely rea-sonable. Some misassignments with regard to previous work [35]were corrected.

We believe that the excited state spin contamination formulapresented here is an important useful aid for conventional TDDFT.It is anticipated that it will be used as an aid to eliminate unphys-ical excited states. However, in at least some cases these statesmay alternatively be corrected to yield physically meaningfulstates [41]. Recently, Wang and Ziegler [44–47] have shown howTDDFT can be extended via a noncollinear reinterpretation of exi-stant functionals to permit spin-flip excitations. (See also earlierwork by Krylov and co-workers [42,43].) Indeed Wang and Zieglerhave obtained similar results with the spin-flip and conventionalTDDFT formalisms for the lower excitation energies of the samemolecules considered in this paper [45], suggesting the validityof the spin-flip alternative. We are optimistic that the spin-con-tamination formula presented in the present work is an importantfirst step towards better TDDFT calculations of excitation spectra inopen-shell molecules and are looking forward to the elaboration ofsystematic methods to reduce and/or eliminate spin-contamina-tion in these excited states via the polarization propagator ap-proach [6], the spin-flip approach [42–47], or still othermethodology [41].

Acknowledgments

This study was carried out in the context of the Groupe deRecherche en Commutateurs Optiques Moléculaires à l’État Solide(COMES), the Groupe de Recherche en Density Functional Theory(DFT), the COST working group D26/0013/02, and a Franco-Mexi-can collaboration financed through ECOS-Nord Action M02P03.Some preliminary results of this study were reported in Ref. [79].FC acknowledges support from the Mexican Ministry of Educationvia a CONACYT (SFERE 2004) scholarship and from the Universidadde las Americas Puebla (UDLAP). AI acknowledges postdoctoralfunding from the Laboratoire d’Études Dynamiques et Structuralesde la Sélectivité (LÉDSS). LJD acknowledges support from the Labo-ratoire de Chimie Théorique of the Département de Chimie Molécu-laire. We thank Pierre Vatton, Denis Charapoff, Régis Gras,Sébastien Morin, and Marie-Louise Dheu-Andries for technical sup-port of the LÉDSS and Centre d’Expérimentation de Calcul Intensif enChimie (CECIC) computers used for the calculations reported here.MEC is grateful to Ed Brothers for implementing our equations ina development version of GAUSSIAN, and for the ensuing discussionsin which he also passed on comments from Mike Frisch. HansJørgen Aa. Jensen is acknowledged for a helpful discussion regard-ing Rowe’s formula. MEC and LJD thank Sébastien Bruneau for use-ful discussions. We also thank the referees of a previous version ofthis paper for suggesting additional references and for identifyingtypographical errors in some of the equations.

Appendix A. Unrelaxed two-electron reduced difference densitymatrix

In Hartree–Fock theory the ground state density matrix is givenby,

C ¼ 12c ^ c; ðA1Þ

where the Grassman product is defined by the doubly antisymmet-rized product,

ðA ^ BÞrq;ps ¼ Ar;pBq;s � Ar;sBq;p � Aq;sBr;p þ Aq;sBr;p: ðA2Þ1 A legal term meaning ‘Let the buyer beware’.

Table 10Example comparison of DhbS2iF with hDbS2i.

Molecule State TDLDA TDLDA TDA

x (eV)aDhbS2i DhbS2iF x (eV) DhbS2i

Stretched H2 1 �1.42 2.47 2.00 1.05 2.00Twisted C2H4 1 �0.74 2.21 2.00 �0.3 2.00BeH 6 5.17 2.0148 1.9722 5.29 1.9797CN 4 7.18 2.0050 1.9852 7.27 1.9838Nþ2 4 8.11 2.0207 1.9971 8.22 1.9965CH2O+ 3b 5.08 2.0503 1.9923 5.26 1.9933CH2O+ 8 8.37 2.0044 1.9801 8.5010 1.9832

a It is common practice in response theory that a negative sign indicates x2 < 0.Thus x ¼ �1:42 eV means x ¼ 1:42i eV. Note, however, that a negative TDA exci-tation energy is real and indicates a de-excitation energy.

b State 4 when the TDA is used.



If we assume that the excited state 2-RDM is also that of a singledeterminant, then

DC ¼ 12ðcþ DcÞ ^ ðcþ DcÞ � 1

2c ^ c ¼ c ^ Dcþ 1

2Dc ^ Dc: ðA3Þ

In fact the unrelaxed TDHF 2-RDDM is given by,

c ^ Dcð Þrq;ps ¼ cr;pDcq;s � cr;sDcq;p � cq;pDcr;s þ cq;sDcr;p;

c ^ DcX;X� �

rq;ps ¼ nrnqnpns

� dr;s

Xa

X�qaXpa � dr;p

Xa

X�qaXsa

"

� dq;s

Xa

X�raXpa þ dq;p

Xa

X�raXsa

#þ nr �nqnp�nsdr;p

Xi

X�isXiq

þ �nrnq�npnsdq;s

Xi

X�ipXir

� nr �nq�npnsdr;s

Xi

X�ipXiq

� �nrnqnp�nsdq;p

Xi

X�isXir;

c ^ DcX;Y� �

rq;ps ¼ 0;

c ^ DcY ;X� �

rq;ps ¼ 0;

c ^ DcY ;Y� �

rq;ps ¼ nrnqnpns

� dr;s

Xa

Y�paYqa � dr;p

Xa

Y�saYqa

"

� dq;s

Xa

Y�paYra þ dq;p

Xa

Y�saYra

#þ nr �nqnp�nsdr;p

Xi

Y�iqYis

þ �nrnq�npnsdq;s

Xi

Y�irY ip

� nr �nq�npnsdr;s

Xi

Y�iqYip

� �nrnqnp�nsdq;p

Xi

Y�irY is:

ðA4Þ

plus a few additional correction terms,

DCXXrq;ps ¼ c ^ Dcð ÞXX

rq;ps

þ nr �nq�npnsX�rpXsq � �nrnq�npnsX

�qpXsr

� nr �nqnp�nsX�rsXpq þ �nrnqnp�nsX

�qsXpr;

DCXYrq;ps ¼ c ^ Dcð ÞXY

rq;ps

þ nrnq�np�ns X�rsYqp � X�qsYrp

�� X�rpYqs þ X�qpYrs

�;

DCYXrq;ps ¼ c ^ Dcð ÞYX

rq;ps

þ �nr �nqnpns Y�pqXsr � Y�prXsq

�� Y�sqXpr þ Y�srXpq

�;

DCYYrq;ps ¼ c ^ Dcð ÞYY

rq;ps

þ �nrnqnp�nsY�prYqs � nr �nqnp�nsY

�pqYrs

� �nrnq�npnsY�srYqp þ nr �nq�npnsY

�sqYrp:

ðA5Þ

Notice how these additional terms also resemble Grassman prod-ucts. This is, of course, required by the doubly antisymmetric ofthe 2-RDDM indices.

References

[1] M.E. Casida, in: D.P. Chong (Ed.), Recent Advances in Density FunctionalMethods: Part I, World Scientific, Singapore, 1995, p. 155.

[2] S. Hirata, M. Head-Gordon, Chem. Phys. Lett. 302 (1999) 375.[3] S. Hirata, M. Head-Gordon, Chem. Phys. Lett. 314 (1999) 291.[4] S. Hirata, T.J. Lee, M. Head-Gordon, J. Chem. Phys. 111 (1999) 8904.[5] J.L. Weisman, M. Head-Gordon, J. Am. Chem. Soc. 123 (2001) 11686.[6] M.E. Casida, J. Chem. Phys. 122 (2005) 054111.[7] D. Maurice, M. Head-Gordon, Int. J. Quantum Chem. Symp. 29 (1995) 361.[8] G.D. Mahan, K.R. Subbaswamy, Local Density Theory of Polarizability, Plenum

Press, New York, 1990.[9] E.K.U. Gross, W. Kohn, Adv. Quantum Chem. 21 (1990) 255.

[10] E.K.U. Gross, C.A. Ullrich, U.J. Gossmann, in: E.K.U. Gross, R.M. Dreizler (Eds.),Density Functional Theory, Plenum Press, New York, 1994, pp. 149–171.

[11] E.K.U. Gross, J.F. Dobson, M. Petersilka, Top. Current Chem. 181 (1996)81–172.

[12] M.E. Casida, in: J.M. Seminario (Ed.), Recent Developments and Applications ofModern Density Functional Theory, Elsevier, Amsterdam, 1996, p. 391.

[13] K. Burke, E.K.U. Gross, in: D. Joubert (Ed.), Density Functionals: Theory andApplications, Springer Lecture Notes in Physics, vol. 500, Springer, Berlin, 1998,pp. 116–146.

[14] R. van Leeuwen, Int. J. Mod. Phys. B 15 (2001) 1969.[15] G. te Velde, F.M. Bickelhaupt, E.J. Baerends, C. Fonseca, C. Guerra, S.J.A. van

Gisbergen, J.G. Snijders, T. Ziegler, J. Comput. Chem. 22 (2001) 931.[16] N.T. Maitra, K. Burke, H. Appel, E.K.U. Gross, R. van Leeuwen, in: K.D. Sen (Ed.),

Reviews in Modern Quantum Chemistry: A Celebration of the Contributions ofR.G. Parr, World Scientific, Singapore, 2002, pp. 1186–1225.

[17] M.E. Casida, in: M.R. Hoffmann, K.G. Dyall (Eds.), Accurate Description of Low-Lying Molecular Excited States and Potential Energy Surfaces, ACS SymposiumSeries 828, ACS Press, Washington, DC, 2002, p. 199.

[18] G. Onida, L. Reining, A. Rubio, Rev. Mod. Phys. 74 (2002) 601.[19] C. Daniel, Coordin. Chem. Rev. 238–239 (2003) 141.[20] M. Marques, E. Gross, in: C. Fiolhais, F. Nogueira, M.A.L. Marques (Eds.), A

Primer in Density Functional Theory, Springer Lecture Notes in Physics, vol.620, Springer, Berlin, 2003, pp. 144–184.

[21] M.A.L. Marques, E.K.U. Gross, Annu. Rev. Phys. Chem. 55 (2004) 427.[22] K. Burke, J. Werschnik, E.K.U. Gross, J. Chem. Phys. 123 (2005) 062206.[23] A. Dreuw, M. Head-Gordon, Chem. Rev. 105 (2005) 4009.[24] M.A.L. Marques, C. Ullrich, F. Nogueira, A. Rubio, E.K.U. Gross (Eds.), Time-

Dependent Density-Functional Theory, Lecture Notes in Physics, vol. 706,Springer, Berlin, 2006.

[25] A. Castro, M.A.L. Marques, H. Appel, M. Oliveira, C.A. Rozzi, X. Andrade, F.Lorenzen, E.K.U. Gross, A. Rubio, Physica Status Solidi 243 (2006) 2465.

[26] P. Elliott, K. Burke, F. Furche, Excited states from time-dependent densityfunctional theory, cond-mat/0703590v1, 22 Mar 2007.

[27] M.E. Casida, in: P. Sautet, R.A. van Santen (Eds.), Computational Methods inCatalysis and Materials Science, Wiley–VCH, Weinheim, 2008, p. 33.

[28] A. Fouqueau, S. Mer, M.E. Casida, L.M.L. Daku, A. Hauser, T. Mineva, F. Neese, J.Chem. Phys. 120 (2004) 9473.

[29] A. Fouqueau, M.E. Casida, L.M.L. Daku, A. Hauser, F. Neese, J. Chem. Phys. 122(2005) 044110.

[30] L.M.L. Daku, A. Vargas, A. Hauser, A. Fouqueau, M.E. Casida, ChemPhysChem 6(2005) 1.

[31] A. Fouqueau Étude théorique de complexes de Fer par la théorie de lafonctionnelle de la densité: Application au problème de piègage de spin, Ph.D.Thesis, Université Joseph Fourier, 3 March, 2005.

[32] S. Zein, S.A. Borshch, P. Fleurat-Lessard, M.E. Casida, H. Chermette, J. Chem.Phys. 126 (2007) 014105.

[33] J. Guan, Ph.D. Thesis, September, 1999.[34] A. Spielfiedel, N.C. Handy, Phys. Chem. Chem. Phys. 1 (1999) 2401.[35] J. Guan, M.E. Casida, D.R. Salahub, J. Mol. Struct. (Theochem) 524 (2000)

229.[36] T. Anduniow, M. Pawlikowski, M.Z. Zgierski, J. Phys. Chem. A 104 (2000) 845.[37] J.G. Radziszewski, M. Gil, A. Gorski, J. Spanget-Larsen, J. Waluk, B. Mroz, J.

Chem. Phys. 115 (2001) 9733.[38] E. Broclawik, T. Borowski, Chem. Phys. Lett. 339 (2001) 433.[39] R. Pou-Amérigo, P.M. Viruela, R. Viruela, M. Rubio, E. Orti, Chem. Phys. Lett.

352 (2002) 491.[40] Z. Rinkevicius, I. Tunell, P. Salek, O. Vahtras, H. Ågren, J. Chem. Phys. 119 (2003)

34.[41] P.T.V. Duijnen, S.N. Greene, N.G.J. Richards, J. Chem. Phys. 127 (2007) 045105.[42] L.V. Slipchenko, A.I. Krylov, J. Chem. Phys. 118 (2003) 6874.[43] Y. Shao, M. Head-Gordon, A.I. Krylov, J. Chem. Phys. 118 (2003) 4807.[44] F. Wang, T. Ziegler, J. Chem. Phys. 121 (2004) 12191.[45] F. Wang, T. Ziegler, J. Chem. Phys. 122 (2005) 074109.[46] F. Wang, T. Ziegler, Int. J. Quantum Chem. 106 (2006) 2545.[47] J. Guan, F. Wang, T. Ziegler, H. Cox, J. Chem. Phys. 125 (2006) 044314.[48] J. Baker, A. Scheiner, J. Andzelm, Chem. Phys. Lett. 216 (1993) 380.[49] J. Wang, A.D. Becke, J.V.H. Smith, J. Chem. Phys. 102 (1995) 3477.[50] F.L. Pilar, Elementary Quantum Chemistry, McGraw-Hill, New York, 1968.[51] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864.[52] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133.[53] J.A. Pople, P.M.W. Gill, N.C. Handy, Int. J. Quantum Chem. 56 (1995) 303.[54] J.M. Wittbrodt, H.B. Schlegel, J. Chem. Phys. 105 (1996) 6574.



[55] A.J. Cohen, D.J. Tozer, N.C. Handy, J. Chem. Phys. 126 (2007) 214104.[56] J.R. Schmidt, N. Shenvi, J.C. Tully, J. Chem. Phys. 129 (2008) 114110.[57] P.-O. Löwdin, Phys. Rev. 97 (1955) 1490.[58] P. Jørgensen, J. Simons, Second Quantization-Based Methods in Quantum

Chemistry, Academic Press, New York, 1981.[59] J. Linderberg, Y. Öhrn, Propagators in Quantum Chemistry, Academic Press,

New York, 1973.[60] D. Lynch, M.F. Herman, D.L. Yeager, Chem. Phys. 64 (1982) 69.[61] P. Cronstand, Z. Rinkevicius, Y. Luo, H. Ågren, J. Chem. Phys. 122 (2005)

224104.[62] F. Furche, R. Ahlrichs, J. Chem. Phys. 117 (2002) 7433.[63] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman,

J.J.A. Montgomery, T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J.Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson,H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T.Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E.K. Andd, H.P.Hratchian, J.B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R.E.Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y.Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S.Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D. Rabuck, K.Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G.B. Andd, S. Clifford, J.Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L.Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, M.Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, J.A.Pople, GAUSSIAN 03, Revision C.02, Gaussian, Inc., Wallingford CT, 2004.

[64] A.M. Köster, P. Calaminici, M.E. Casida, R. Flores, G. Geudtner, A. Goursot, T.Heine, A. Ipatov, F. Janetzko, S. Patchkovskii, J.U. Reveles, A. Vela, D.R. Salahub,DEMON2K, Version 1.8, The deMon Developers, 2005.

[65] R.E. Stratmann, G.E. Scuseria, M. Frisch, J. Chem. Phys. 109 (1998) 8218.[66] A. Ipatov, A. Fouqueau, C.P. del Valle, F. Cordova, M.E. Casida, A.M. Köster, A.

Vela, C.J. Jamorski, J. Mol. Struct. (Theochem) 762 (2006) 179.[67] A.J. Sadlej, Collec. Czech. Chem. Commun. 53 (1988) 1995.[68] A.J. Sadlej, Theor. Chim. Acta 79 (1992) 123.[69] A.J. Sadlej, Theor. Chim. Acta 81 (1992) 45.[70] A.J. Sadlej, Theor. Chim. Acta 81 (1992) 339.[71] D. Maurice, M. Head-Gordon, J. Phys. Chem. 100 (1996) 6131.[72] G. Das, T. Janis, A. Wahl, J. Chem. Phys. 61 (1974) 1274.[73] H. Nakatsuji, Chem. Phys. Lett. 177 (1991) 331.[74] N. Kosugi, H. Kuroda, S. Iwata, Chem. Phys. 39 (1979) 337.[75] P.J. Bruna, M.R.J. Hachey, F. Grein, Mol. Phys. 94 (1998) 917.[76] A.O. Bawagan, C.E. Brion, E.R. Davidson, C. Boyle, R.F. Frey, Chem. Phys. 128

(1988) 439.[77] M.E. Casida, C. Jamorski, K.C. Casida, D.R. Salahub, J. Chem. Phys. 108 (1998)

4439.[78] E. Tapavicza, I. Tavernelli, U. Rothlisberger, C. Filippi, M.E. Casida, J. Chem.

Phys. 129 (2008) 124108.[79] M.E. Casida, A. Ipatov, F. Cordova, in: M.A.L. Marques, C. Ullrich, F. Nogueira, A.

Rubio, E.K.U. Gross (Eds.), Time-Dependent Density-Functional Theory,Springer, Berlin, 2006, p. 243.


Excited-state spin-contamination in time-dependent density-functional theory for molecules with...

Documents

Transcript of Excited-state spin-contamination in time-dependent density-functional theory for molecules with...