12 January 1996
CHEMICAL
PHYSICS
LETTERS
ELSEVIER
Chemical Physics Letters 248 (1996) 189-198
Simplified methods for equation-of-motion coupled-cluster
excited state calculations
Steven R. Gwaltney, Marcel Nooijen, Rodney J. Bartlett
Quantum Theory Project, University of Florida, Gainesville, FL 32611-8435, USA
Received 5 September 1995; in final form 9 November 1995
Abstract
Simplified equation-of-motion coupled-cluster (EOM-CC) methods derived from matrix partitioning and perturbation
approximations are presented and applied to a variety of molecules. By combining a partitioned EOM-CC method with an
MBPT(2) treatment of the ground state, we obtain an iterative n s method which gives excitation energies that normally fall
within 0.2 eV of the full EOM-CCSD excitation energy. Results are shown to be superior to other simplified approaches that
have been proposed.
I. Introduction
The equation-of-motion coupled-cluster (EOM-CC) method [1-4] is a conceptually single reference, generally applicable, unambiguous approach for the description of excited [2-4], electron-attached [5] or ionized
states (see Ref. [6] for a review). All follow from simple consideration of the SchriSdinger equation for two
states, a reference state 0 (not necessarily the ground state), and an excited (electron attached or ionized) state K.
Considering H to be in second quantization, where the number of particles is irrelevant, we have
/4~o = Eo~o,
( 1)
H~K = E K ~ .
(2)
We then choose to represent the excited state eigenfunction as
=
(3)
from which we readily obtain
[ H, g ~ ] ~0 = o J ~ / ~ 0
(4)
for ~oK = E~ - E 0, from subtraction of Eq. (1) from Eq. (2) after left multiplication by £2~.
The choice of ~K defines the particular EOM with i, j, k . . . . indicating occupied orbital indices and
operators, while a, b, c . . . are unoccupied orbitals and operators. Also, p, q, r . . . refer to orbitals and
operators of either occupation. For electronic excited states,
0009-2614/96/$12.00 © 1996 Elsevier Science B.V. All rights reserved
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S.R. Gwahney et a l . / Chemical Physics Letters 248 (1996) 189-198
190
(5)
i,a
i>j
a>b
for electron attachment
~)~EA='4K:EAa(K)at+
a
E A~b(K)atJ~*+
a>b,j
E
"ajk--~bc'i,K)atjbtk~t+....
(6)
j>k
a>b>c
and for ionization
g)2 = / ~ =
EIi(K) ~+ E li~(K)~atf+ E I~jbktatJ~*k + . . . .
i
i>j,a
(7)
i>j>k
a>b
Coupled-cluster theory is introduced by choosing
~0 = e x p ( T ) • o ,
(8)
where 7~ is the usual excitation operator, and • 0 represents some independent particle model reference. Since
[ ~ , T] = 0 for any of the above choices, we can commute the operators to give
[H,
~ ] O 0 = (HO)cO0 = toKYOo,
(9)
where
/~ = exp( - T) H e x p ( T )
(10)
and ( H O ) c indicates the open, connected (contracted) terms that remain in the commutator. In this way, all the
ground state CC information is contained in H, now generalized to have three- and four-body terms [4]. Note,
is also non-Hermitian necessitating that both its left ((7 K) and right (C K) eigenvectors, which form a
biorthogonal set, ( C x C ~ = tSx_~), be considered for a treatment of properties. In matrix form,
( H C , ) c = C K t o K,
C2H = C2 toK,
(11)
where C K = R K, A K or I K depending on the process. Consequently, EOM-CC reduces to a CI-like equation that
provides the relevant excitation energies directly.
Besides the obvious approximations, such as T = T 1 + T 2 and OK being limited to single and double
excitations, which defines EOM-CCSD, and various triple excitation extensions, EOM-CCSD(T) [7], EOMC C S D T - 1 and E O M - C C S D T [8]; one can conceive of many other approximations to the basic EOM-CC
structure. Instead of Eq. (8), for the reference state a perturbation approximation might be chosen, such as
~0O~)= (1 + T (') +
1T(1)T(')+
T C2) + ... ) O o,
(12)
where one truncates at a particular order, m. (See Ref. [9] for the CC, non-Hartree-Fock definition of the
various orders.) Alternatively, rather than retaining the full perturbation approximation to ~0(m), we could
truncate H itself to some order [10,11] ]
We can also conceive of perturbative approximations on ~K, too. The latter are, perhaps, most easily viewed
from the partitioning approach to perturbation theory [12]. That is, Eq. (12) can be partitioned into the spaces P
I Stanton and Gauss use the name EOM-CCSD(2) to refer to what we are calling EOM-MBPT(2) [10].
S.R. Gwaltneyet al. / ChemicalPhysicsLetters 248 (1996) 189-198
191
and Q, where P represents the principal configuration space (of dimension p) and Q (of dimension q)
represents its orthogonal complement. Then it is well known that we can consider an effective Hamiltonian,
(13)
whose eigenvectors are solely defined in the P space
(14)
Hpe ce = ce t°K
for the first several eigenvalues, toK. Expanding the inverse in Eq. (13) provides a series of perturbative
approximations to Hee, or for the eigenvectors ~'e in Eq. (14) [12]. With P chosen to be the space of single
excitations, and Q that of double excitations, such partitioned EOM-CCSD results, first presented in 1989 [2],
have been shown to retain most of the accuracy of the full EOM-CCSD method. Other approximations can be
made. To introduce triple and quadruple excitations, selective double excitations could be retained in P and at
least diagonal approximations could be made for the triple and quadruple excitation blocks of HQQ.
The objective of this Letter is to reconsider such partitioned and perturbation-based approximations to the full
EOM-CCSD method. We will demonstrate that excellent accuracy may be obtained within a much less
2 4v oc n 6, for
expensive computational structure. In particular, whereas the full EOM-CCSD is proportional to non
3
n
5
n basis functions, we will develop a purely n2nv ¢x
procedure that shows great promise for large molecules.
2.
Theory
2,1. Partitioned EOM-CC for excited states
In a typical EOM-CCSD calculation, the excitation energy and excited state properties are calculated via
diagonalization of the non-symmetric matrix [4],
= L Ds
'
where Hss stands for the singles-singles block of the matrix, etc. The major step in an iterative diagonalization
[13] of the matrix is multiplying the matrix by a trial vector C. In the EE case, the equations for the
multiplication are as follows [4,8] 2:
[HssC]~ = ~-'.FaeC ~ - ~F,~iC,~ + EWamie Ce ,
e
[HsDC]
m
(16)
em
a
I
ef
I
i = E F m e C a e + -2 E W a m e f C i r a -- ~ E WmnieCaen,
em
reef
rune
[RosC]i ab = P ( a b ) EWmaijCb~ + P ( i j )
m
E
Wbmfe Ce
EWabejC
e
--
(17)
7
P(ij) ~(~e
t~b'
18)
In Refs. [4,8], the three body terms, the last two terms in Eqs. (18) and (19), were incorrectly written as including bare two electron
integrals. The method, as implemented,has always correctly used H elements.
S.R. Gwaltney et a l./Chemical Physics Letters 248 (1996) 189-198
192
ae
[HDDC] ab
ij = P ( ab ) E FbeCij -- P ( ij)
e
m
~.,FmjCi~+ ~1 E
ef
WabefCij
ef
+ ~1 EWm.ijcab. + P ( a b ) P ( i j ) EWb.,j.Cimae
mtl
em
- ½ P ( a b ) ~f (m~nneW.mfeCe~)t fb +½P(ij)~n ( ~--~""mfe'~im]'J'"
~
¢'~fel,ab
mef
(19)
The permutation operator P(qr) is defined as
P( q r ) • ( . . . q . . ,
r . . . ) = ¢( . . . q . . . r . . .
) - ~(...r...q...
).
(20)
Fpq and Wpqrs are
presented elsewhere_ [4].
In the partitioned scheme, HDD is replaced by HODD, where H 0 is the usual M011er-Plesset unperturbed
Hamiltonian from many-body perturbation theory [6]. In other words, the unfolded matrix analogous to Eq. (13)
to be diagonalized is approximated by
- ( ss Rso )
H=~fio s
(21)
Hood ,
and Eq. (19) becomes
Ehec,7 -
[H 0 DDC ]isb = P(ab)
e
P(ij)
Y'.fm:c,~.
(22)
m
Here, fpq is the Fock operator. In the case when the reference function is composed of canonical or
semi-canonical Hartree-Fock orbitals, then H 0 DD becomes diagonal with differences of orbital energies on the
diagonal. From examining the equations it is clear that the method is formally an iterative n 5 method, and the
results are invariant to rotations among occupied or unoccupied orbitals. This partitioning scheme differs from
that presented in Ref. [2] in two ways. Geertsen et al. included the full H elements on the diagonal instead of
just including H o elements in the doubles-doubles block. Also, Geertsen et al. did not include the three body
terms (the last two terms in Eq. (18)) in their work.
Instead of partitioning Eq. (21) to the form of Eq. (13), we iteratively diagonalize Eq. (21) to obtain
solutions. Since a matrix the size of the singles plus doubles is still being diagonalized, the eigenvectors
correspond to all possible single excitations plus all possible double excitations, just as in a full EOM-CCSD
calculation. Therefore, the same techniques used to calculate the properties of a full EOM-CCSD wavefunction
[4] can be used to calculate properties of the partitioned EOM-CCSD (i.e. P-EOM-CCSD) wavefunction.
Since H is not Hermitian, its left and right hand eigenvalues are the same, but the eigenvectors differ [4]. The
equations for the left-hand side are similar to those presented above.
2.2. MBPT[2] ground state
The first logical perturbative approximation to
= e- rHer= ( He r )c
=
EFpq{p*q}+
E Wpqrs{Ptqtsr} + higher order terms
pq
pqrs
(23)
is obtained by keeping only terms through second order. Such an approximation defines an EOM-MBPT[2]
method for excited states (i.e. an EOM-CC calculation based on a MBPT[2] ground state instead of a
S.R. Gwaltney et a l . / Chemical Physics Letters 248 (1996) 1 8 9 - 1 9 8
193
coupled-cluster ground state) [11]. If we insist upon a method correct through second-order for the non
Hartree-Fock case, the explicit equations for ~tz) are
(24)
F., = 0,
F~b =f~b + ~-~tamfmb+ ~,t/.(mall 3"o) -- ½ ~, t~,~.(mnll be),
m
fen
Fij = f / j + Y'.t~fie + Y'~te(imll je) - ½ ~_,t~,~(imll ef),
e
(25)
emn
em
(26)
efrn
Fi. =f/. + Y'.t~,(imll ae),
(27)
em
(28)
W a b i j = O,
Wij, t = (ij[I kl) + P( kl) Y'~t~( ijll ke) + ½ ~,t~f( ijll ef),
e
(29)
ef
ab
W~bcd = (abll cd) - P( ab) ~tbm(amll cd) + ~I ~tm.(mnll
cd),
m
(30)
rtltt
W~ib¢ = (ai II be> - ~,t~,(mi II bc),
(31)
m
Wij,~ = <jk II ia) + ~_~t~ ( jk II ea),
(32)
e
(33)
W~job = <ijll ab ) ,
W~ijb = (ai II jb) + ~ tf(ai II eb) - ~ t~,(mi II jb> - ~ t~,~(mi II eb),
e
m
(34)
em
W~bc~= (abll ci) - P ( a b ) EtT~(amll ce) + i y,t~b(ci II mn)
era
ab
a
ti~fm~
+ Y'~tie (abll ce) -- P(ab) ~_.tm(mbll
ci),
+E
m
Wiajk =
e
(35)
rtl
<iall jk) - P( ij) Y'.t~(im II je) + ½ ~.,tff(iall ef)
era
ef
+ Y'.tf;f/, + Y'.t~,(imll jk) - P ( i j ) Et;(iall ek),
e
m
(36)
e
In all of these equations t~ and t~b refer to their values only through first order. That is, t! q~ = f/a//(E" i -- ~a) and
= (abll i j ) / / ( ~ i + ej -- e a -- Eb).
Since approximating the excited state with a partitioned EOM-CC calculation and approximating the ground
state with a MBPT(2) calculation are independent approximations, we can obviously combine them into a
partitioned EOM-MBPT(2) (P-EOM-MBPT(2)) method. When combined, the H elements Wijkt, Wii~b, and the
most numerous, W~bcd, are not needed. Therefore, in the Hartree-Fock case, the ( a b II cd) integrals never need
to be calculated. The (ab II cd> integrals would contribute to the W~bci H element multiplied by a T}l], but in
the Hartree-Fock case all T~l] are zero. A few n 6 terms still remains in the calculation of the H elements, but
these terms only need to be calculated once. In most calculations the cost of the iterative n s step in the excited
state calculation will dominate over the n 6 s t e p involved in calculating the H elements.
t!) lab
S.R. Gwaltney et a l . / Chemical Physics Letters 248 (1996) 189-198
194
3. C a l e u l a t i o n s
3.1. Be atom
Table 1 lists energies calculated using the four methods described above (EOM-CCSD, P-EOM-CCSD,
EOM-MBPT(2), and P-EOM-MBPT(2)) for the first few singlet excited states in beryllium. The basis set from
Ref. [14] was used in the calculations. From the mean absolute errors, it would appear that only the EOM-CCSD
calculation is able to reproduce the full-CI excitation energies (which are exact within the basis set used).
However, if the 1 ~D state is excluded from the determination of the mean absolute error, the errors become
0.010 eV for EOM-CCSD, 0.151 eV for P-EOM-CCSD, 0.431 eV for EOM-MBPT(2), and 0.300 eV for
P-EOM-MBPT(2). With an AEL value of 1.60, the 1 1D state is dominated by double excitation character. The
AEL is a measure of the number of electrons excited in an excitation [4]. A value of 1.00 is a pure single
excitation, while 2.00 corresponds to a pure double excitation. Since the doubles-doubles block of the EOM
wavefunction is approximated in this partitioned scheme, it is reasonable to expect any state with appreciable
double excitation character to be poorly described by this partitioned EOM calculation.
Even with the 1 1D excluded, the calculations based on a MBPT(2) ground state are still very poor. This poor
behavior can be ascribed to the inadequacy of the MBPT(2) wavefunction in describing the ground state of Be.
In a CCSD calculation the largest T2 amplitude is 0.065, while the largest T2tjl amplitude in the MBPT(2)
calculation is only 0.034. Also, the MBPT(2) calculation only obtains 73% of the correlation energy recovered
by the CCSD calculation. Clearly, MBPT(2) is not adequate to describe the ground state, and any excited state
calculation based on that MBPT(2) ground state will suffer accordingly.
In Table 2 the first few triplet excited states of beryllium starting from the singlet ground state are presented.
Since all of the calculated states are single excitations, the EOM-CCSD method does exceedingly well, while
the P-EOM-CCSD also does a good job of describing the states.
Table 1
Be singlet excitationenergies (in eV)
State
AEL a
EOMCCSD
P-EOMCCSD h
1 ip
2 lS
1 ~D
2 ip
2 iD
3 ~S
3 tp
3 tD
4 IS
4 ip
5.323
6.773
7.139
7.468
8.055
8.084
8.309
8.548
8.583 d
8.700
5.666
6.985
10.579
7.653
7.892
8.203
8.432
8.548
8.694
8.792
0.014
(0.010) e
0.485
(0.151) e
mean abs.
error
1.07
1.06
1.60
1.06
1.21
1.04
1.05
1.09
1.04
1.04
EOMMBPT(2)
4.869
6.329
6.682
7.023
7.618
7.648
7.873
8.115
8.267
0.428
(0.431) e
P-EOMMBPT(2)
FulI-CI ¢
5.228
6.576
10.170
7.224
7.477
7.788
8.011
8.132
8.278
8.374
5.318
6.765
7.089
7.462
8.034
8.076
8.302
8.536
8.600
8.693
0.578
(0.300) e
a Ref. [8]. The AEL is for the EOM-CCSD wavefunction.
b Because of a different implementationof the partitioning,these numbers are slightly differentthan those in Ref. [2].
c Ref. [14].. d Ref. [8].
e Average error without l i d double excited state.
S.R. Gwalmey et al./ Chemical Physics Letters 248 (1996) 189-198
195
3.2. E x a m p l e molecules
While comparisons with full-CI calculations are useful, it is also informative to look at the performance of
the methods for more c o m m o n molecules. These calculations will also provide an opportunity to compare our
methods with other single reference methods used today. Table 3 presents calculations on four molecules:
formaldehyde, ethylene, acetaldehyde and butadiene. The calculations are performed at the M P 2 / 6 - 3 1 G "
geometries given in Refs. [15-17]. A 6-311(2 + , 2 + )G * " basis [15] is used for formaldehyde and ethylene,
with a 6 - 3 1 1 ( 2 + ) G " [16] basis for acetaldehyde and butadiene. All electrons are correlated except for
butadiene, where the first four core orbitals are dropped.
All of the methods presented, except for the CIS-MP2 [18], can be viewed as approximations to the full
E O M - C C S D method. T D A (or CIS) [19] is the crudest o f the methods in that the excited state is given as a
linear combination o f single excitations out of a single reference ground state. This method includes no dynamic
correlation. CIS(D) [20] provides a non-iterative n 5 perturbation correction to the CIS energy. CIS-MP2 also
provides a correction to the CIS energy, but the method is not size-consistent and scales as n 6 [20]. The
partitioned methods provide an iterative n 5 excited state method based on either a n 5 ground state (P-EOMMBPT(2)) or an iterative n 6 ground state (P-EOM-CCSD). The full E O M - C C S D method is an iterative n 6
method. The partitioned methods, along with T D A and E O M - C C S D , have the advantage of providing a
wavefunction for calculating properties, instead of just an energy.
Our assignment of states agrees with the assignment of states by Wiberg et al. and with the corrected
assignment o f Head-Gordon et al. [21], except for two differences. The states have been ordered based on their
experimental excitation energies, or, where not available, their E O M - C C S D excitation energies, instead of their
CIS excitation energies. Also, Head-Gordon et al. [20] had incorrectly assigned the IA~ E O M - C C S D state at
9.27 eV in formaldehyde to the valence 4 1A 1 state. Based upon the state's properties, including an ( r 2 ) value
o f 54.4 au 2 (compared to the ground state ( r 2) value of 20.5 au2), we have reassigned the 1A 1 E O M - C C S D
state to the Rydberg 3 1A~ state. The 4 ~A~ state (the ~r" ~ w state), with an ( r 2) value o f 32.6 au 2, has an
E O M - C C S D excitation energy o f 10.00 eV.
Considering the basis sets and the relatively inexpensive methods used here, these calculations are not meant
to be definitive. For previous results on these molecules see, for example, Refs. [22,23] for formaldehyde. See
also the references in Refs. [15-17]. Instead, the mean absolute errors o f the methods compared to experiment
and compared to E O M - C C S D will be used to access the quality of the methods. The mean absolute error for
Table 2
Be triplet excitation energies (in eV)
State
AEL a
EOMCCSD
P-EOMCCSD b
EOMMBPT(2)
P-EOMMBPT(2)
FulI-CI ¢
1 3p
2 3S
2 ~P
1 3D
3 3S
4 3p
2 3D
4 3S
5 3p
2.729
6.447
7.301
7.748
7.991
8.278
8.456
8.58 a
8.70 d
2.819
6.583
7.424
7.866
8.089
8.366
8.539
8.647
8.766
2.271
5.997
6.868
7.316
7.557
2.366
6.140
7.006
7.448
7.668
7.949
8.121
8.228
8.348
2.733
6.444
7.295
7.741
7.985
8.272
8.449
8.560
8.686
0.008
O.104
0.436
mean abs.
error
1.01
1.04
1.04
1.04
1.04
1.03
1.03
a The AEL is for the EOM-CCSD wavefunction.
b See footnote b, Table 1. c Ref. [14]. 0 Ref. [3].
8.025
0.321
196
S.R. Gwaltney et al. / Chemical Physics Letters 248 (1996) 189-198
Table 3
Example molecules (energies in eV)
Molecule
State
CIS ~
CISMP2 ~
CH20
1
1
2
2
2
3
1
3
4
4
tA2(V)
i B2(R )
I B2(R)
lAl(R)
IAz(R)
I Bz(R)
JBI(V)
tAt(R)
I B2(R)
~A~(V)
4.48
8.63
9.36
9.66
9.78
10.61
9.66
10.88
10.86
9.45
C2H4
1
1
1
1
2
2
1
3
2
2
t B3u(R)
~Bi,(V)
I B ~(R)
IB2g(R)
lAB(R)
I B3u(R)
IAu(R)
SB3u(R)
I BLu(R)
I Btg(R)
1
2
3
2
4
5
6
3
4
7
I
1
1
2
2
2
2
3
4
3
3
3
4
C2H40
C4H6
Ref. [15-17].
CIS(D) t,
P-EOMMBPT(2) c
P-EOMCCSD c
EOMCCSD
4.58
6.85
7.66
8.47
7.83
8.46
9.97
8.75
8.94
9.19
3.98
6.44
7.26
8.12
7.50
8.21
9.37
8.52
8.63
8.80
4.31
7.10
7.97
8.02
8.25
8.99
9.61
9.24
9.39
10.08
4.41
7.10
7.95
8.01
8.23
8.97
9.68
9.21
9.38
10.24
3.95
7.06
7.89
8.00
8.23
9.07
9.26
9.27
9.40
10.00
t,
b
b
b
b
t,
b
c
b
c
4.07
7.11
7.97
8.14
8.37
8.88
7.13
7.74
7.71
7.86
8.09
8.63
8.77
8.93
9.09
9.09
7.52
8.39
8.14
8.12
8.42
8.92
9.00
9.14
9.38
9.31
7.21
8.04
7.84
7.86
8.18
8.69
8.80
8.96
9.18
9.12
7.45
8.20
8.08
8.12
8.43
8.95
9.07
9.23
9.42
9.42
7.51
8.36
8.14
8.18
8.48
9.00
9.12
9.28
9.51
9.48
7.31
8.14
7.96
7.99
8.34
8.86
9.01
9.18
9.39
9.38
t,
b
b
b
b
b
b
b
c
~
7.11
7.60
7.80
8.01
8.29
8.62
Ig'(V)
~/((R)
t,~(R)
IK'(R)
~K(R)
~K(R)
I,~(R)
l,n/'(R)
I,~'(V)
I~(V)
4.89
8.51
9.22
9.37
9.30
10.19
10.26
10.31
9.78
9.73
5.27
6.71
7.57
7.37
8.00
8.09
8.08
8.10
10.34
9.07
4.28
6.13
7.04
6.90
7.42
7.70
7.70
7.74
9.34
8.50
4.65
6.88
7.57
7.70
7.77
8.42
8.53
8.58
9.61
9.58
" 4.71
6.84
7.52
7.64
7.70
8.35
8.47
8.51
9.65
10.07
4.26
6.78
7.49
7.64
7.68
8.39
8.51
8.57
9.23
9.44
b
b
b
b
b
b
b
b
¢
~
iBu(V )
f Bg(R)
IA,(R)
rA,,(R)
IBu(R)
IBg(R)
tAg(R)
I Bg(R)
I Bg(R)
IAg(R)
~Bu(R)
tA u(R)
IAu(R)
6.21
6.11
6.45
6.61
6.99
7.22
7.19
7.25
7.39
7.45
8.05
7.78
7.92
7.00
6.73
7.03
7.11
7.58
7.66
7.74
7.74
7.87
7.88
8.40
6.75
7.66
6.29
6.11
6.44
6.55
7.03
7.17
7.19
7.24
7.40
7.44
8.01
7.73
7.86
6.52
6.40
6.72
6.85
7.29
7.47
7.43
7.54
7.70
7.74
8.32
8.07
8.19
6.63
6.39
6.73
6.84
7.34
7.47
7.44
7.53
7.69
7.73
8.31
8.07
8.18
6.42
6.20
6.53
6.67
7.17
7.31
7.10
7.39
7.55
7.61
8.21
7.92
8.06
b
b
b
b
b
b
b
t,
b
b
c
¢
c
b Ref. [20].
9.33
9.34
4.28
6.82
7.46
7.75
8.43
8.69
5.91
6.22
6.66
7.07
7.36
7.4
7.62
7.72
8.00
8.18
8.21
c Present work.
each of the methods
w i t h r e s p e c t to t h e e x p e r i m e n t a l
0.32 eV for CIS(D),
0.17 eV for P-EOM-MBPT(2),
EOM-CCSD
Compared
method.
Exp. ~
to t h e m o r e c o m p l e t e
v a l u e s g i v e n is 0 . 6 6 e V f o r C I S , 0 . 4 1 e V f o r C I S - M P 2 ,
0.20 eV for P-EOM-CCSD,
EOM-CCSD
method,
and 0.14 eV for the full
the mean absolute errors of the
S.R. Gwaltney et al. / Chemical Physics Letters 248 (1996) 189-198
197
various methods are 0.70 eV for CIS, 0.39 eV for CIS-MP2, 0.35 for CIS(D), 0.12 eV for P-EOM-MBPT(2),
and 0.16 eV for P-EOM-CCSD.
As should be expected from its simple nature, TDA performs poorly. The average errors for CIS-MP2 and
CIS(D) are similar, although, as previously noted [20], the CIS-MP2 energies are more erratic. The P-EOMMBPT(2) and P-EOM-CCSD energies are also similar, seldom differing by 0.1 eV. This suggests that for these
cases where MBPT(2) is able to well represent the ground state, P-EOM-MBPT(2) should be able to well
describe singly excited states. A balance argument between the ground and excited state would also tend to
favor the P-EOM-MBPT(2) method, since the partitioning, as discussed above, can be viewed as a second-order
in H perturbation expansion for the excited state.
Looking only at the valence states, as designated by Wiberg et al. [15-17], the mean absolute errors from the
EOM-CCSD energies are 0.45 eV for CIS, 0.68 eV for CIS-MP2, 0.33 eV for CIS(D), 0.23 for P-EOM-MBPT(2),
and 0.38 eV for P-EOM-CCSD. This is compared to the Rydberg states, where the mean absolute errors from
the EOM-CCSD energies are 0.76 eV for CIS, 0.32 eV for CIS-MP2, 0.35 eV for CIS(D), 0.10 for
P-EOM-MBPT(2), and 0.11 eV for P-EOM-CCSD. It appears that the partitioned methods on average do not
describe valence states as well as they describe Rydberg states. Since orbital relaxation is often greater for
valence states, it is possible the reduced relaxation available by approximating the doubles vector might cause
larger errors for valence states.
These calculations also give a good chance to measure the time savings of the partitioned method. For
example, for butadiene the full EOM-CCSD calculation took 25905 s on an IBM R S / 6 0 0 0 model 590 to
calculate eighteen excited states (both right and left hand sides). To calculate the same states with the
P-EOM-CCSD method took 3998 s. These times only include the time for the excited states. P-EOM-MBPT(2)
also saves time in the calculation of the ground state and in the formation of H, as well as saves disk space.
4. Conclusions
In this Letter a formalism is presented and results are given for partitioned methods based on the
equation-of-motion coupled-cluster theory, where the ground state can be described by either a CCSD or an
MBPT(2) wavefunction. The partitioned methods provide an iterative n 5 method (plus a n n 6 step for forming
elements) for excited states. When the ground state of the system is well described by an MBPT(2)
wavefunction, the P-EOM-MBPT(2) method provides an inexpensive way to accurately calculate the energies
and properties of singly excited states. For systems less well described by a MBPT(2) wavefunction, the
P-EOM-CCSD method is a generally accurate, but more economical, alternative to a full EOM-CCSD
calculation. P-EOM-MPBT(2) is a superior n 5 method to CIS(D).
Acknowledgement
This work has been supported by AFOSR grant F49620-95-1-0130, by AFOSR AASERT grant F49620-95-I0421, and by a NSF Fellowship for Graduate Study for SRG.
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