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Application of perturbation theory in large configuration interaction calculations
VO!UIIE 37;number
1
1 January
CHEMICAL PHYS!CS LETTERS
1976
:
APPLICATION OF PELZTURBATION THEORY IN LARGE CONFIGURATION !NTERACTION CALCULATIONS+ PJ.
FORTUNE
and Bruce J. ROSENBE.RG
Chemistry Division, Argotwe
National Laboratory.
Argome.
Illinois
60439.
USA
Received 2s JUy 1975 Revised manuscript received 20 August 1975
Ray!aigh-Schrijdinger Perturbation theory has been applied through fifth order in the energy, to the problem of estimating the rodts of the secular equation in large configuration interaction calculations. The NO:, 03 and Hz0 molecules are used as test uses, wit% accuracy as good as O.Gl eV, with appropriate choice of zero order problem.
1. Introduction The principal technicye used to include electron correlation effects in theoretical molecular calculations is
test calculations. The results of the test calculations performed in this work further indicate that, rvith an appropriate defini-
the configuration interaction (C!) form of the linear variational me*Lhod. The number of configurations in a given czilcula t-IOII depends on t,‘le level of accuracy with which one is concerned. It is not uncommon for
tion of the zero order problem, second order perturbation theory can be sufficient for some applications. For such cases, accuracy on the order of a millihartree can be achieved while constructing only part of the full Hmatrix.
this number to reach several thousand, so that application of configuration selection techniques to minimize the size of the secular determinant prior to diqonaliza-
2. Theory
tion can be important. Similarly, it would be desirable to be able to obtain reliable estimates of the roots of a large CI matrix without using (or therefore construct-
ing) the entire matrk. In this work we have appliad
Rayleigh-Schredinger perturbation theory (RSFT) [l] through fifth order to the matrix eigenvalue problem and expressed the resulting total energy in terms of a sum of en&y contributions from individual confi;urations. Tnese energy contributions can be used to eliininate a subset of con-
figurations from the original
CI list. For large matrices,
the method
requires a fraction of the time needed for diagonalization via the usual iechtiques [2] , with the difference between the perturbation results and those from the full diagonalization being quite small in our
The application of RSPT to the problem of determining energy contributions of individual configurations is natural. The problem at hand is one of solving the secular equation: Hc = EC.
(1)
To
apply RSPT, we separate the H-matrix into a sum of zero order plus pertllrbing terms: H= Ho +AV.
(3
Here, X is a formal ordering parameter which, in the fki analysis, will be set to one. With the diiension of the H-matrix denoted by N, we take the form of HO and
v to be:.
* Based on work performed unde; the auspices of the Uaited ,Setes Energy Resev&.and Davelopmen! Administration. .. .
.llO’.,._ ,:,-_
: -. ‘. _._ ..-:..y
‘.:. .,: .,:_.-
-y.,
_’
:
y,. ..:
,, .,
,.
.-.,
.
.;
1
;
: _..’
Volume
cmwa_
37, number 1
HI7 ...HIK 4, ._ !G __.H,K Hll
_d
.
.
.
.
0
_.. 0
0
. .. 0
._
..-;
(HK1HK2...~;
Ho=
,jQ:
0;
:
yY’Tl,K+l,‘-‘;
PHYsIcs LETTERS
I January 1976
1
(3)
i
,
and V=H-HO. The zero
(4)
order
problem
is then
one of diagonahzing
the (relatively small) K X K submatrix of H, with Jth eigenvalue denoted by ei”), and eigenvectors represented by:
Note that in the expression for @J, the last two bracketed terms give rise to equal sums; however, for the purof defining energy contributions, bine them. Defming
pose
we do not com-
00)
(5)
we write the energy contribution S,, of the r,th configuration, order by order, as follows:
and c$o’ = [0 . ..O....
l,...OIT
J>K.
(6)
In what follows, we assume that the configuration list is ordered so that the first K terms correspond to the dominant set. Ways in which this set can be chosen will be discussed in a later section of this paper. For the sake of simplicity, be the lowest
we take
the root
of interest
to
one.
With the definition that the derivative of a vector is the vector of the derivatives of the individual components, one easily- obtains the usual perturbation equations [ I] _One can expand the solutions to these equations in terms of the zero order vectors, and defining (7) and
it can be shown that the expressions for the first five corrections to the lowest root are [l] :
(11) The-total lowest
contribution root
s,r= i:
is then
of the rzth configuration defined
to the
by:
s,‘:“.
111= I Using eqs. (3) and C4), it is straightfo~vard the various matrix (11) are given by: ‘Jr,=
’
products
to show
that
P’,, which enter eqs. (7)-
(i3n) R Cl3b)
=HJIC1 -S/r)
3. Calculation
J>K.I>K.
(13c)
and discussion
For the purposes
of testing
the efficacy
of the above
i
-: Vulumc 37. notiber :
CIIEWCAL
i Jenwry
PHYSICS LETTERS
1976
Tab!.e 1
.. Results of perrurbation
calculations
State
Molecule
___._...-,_
_
eqs. (7)-(13)
using IF)
.-..
.._..
.-.... .
Nb) Epert .. . . .-.- .- --. . ..- ..-..I.---
lA&Vlid~
‘diag
. NO;
X!Al
1
324
-203.6683
-103.7829
3.111
NO;
X’A,
-203.7825 -224.6105
-224.4363
O.Oi 1 4.194
9s .HzO
X’A, X’AI X’AI X’AI
224 198
-203.7829
03
33 1 2 24 1
198 198 371
-224.4463 -224.4555’ -76.1126
-224.4563 -224:456;3 -76.1114
0.272 0.022 0.033
H,O -~--
X’A, -... -__-.._-._
1
323sc)
-76.3009
-76.2974
0.095
03.
of
cf
contigusatiols.
Calculation includes singe and double excitations d, 1 eV= 23.2558 kcal/mole.
Table 2 Order-by-order
perturbation
enxgies
03
--.-----_-...--
only. Present program limitations
estension
of this we
--
eS3)
$)
$)
-203.76597
-0.01753
0.00226
-0.00219
b.00094
-224.43616 .---..-..--.
-0.01997
0.00204
-0.00x7
0.00111
-._...- ---------_.-._--__.
.--...--. ---
H20, NO; and O3 molecules are summarized in tabie 1. For the NO; and 03 calculations, the convergence
of the perturbation
expansion is ~Iustrated in table 2. 3 lists the dependence of the perturbation results on K for the H,O calcu!ations. The results listed in table 1 indicate that the NO; and 0; problems with K = 1 are the most poorly conditioned cases of those which we considered. This is also manifested by the fact that the squares of the dom-
.__.____-____-__________
Table 3 Perturbation results as a function of K for the Iv = 371 [iz(.j CI calculations”) _-_.._..., ----__-_------
---.---7-s---.
Tabk
inant C’Imiving coefficient respectively,
for these cases are 0.8579 while
that
to K > 1.
---___
$)
method, we have applied it to several cases of chemical interest. These calculations, on the X1 Al states of the
and 0.8225,
prohibit
for NOf (K = 33) and 03 (K = 24) caIc&tions
$)
+ N32
-----
of zero order problem.
‘)K. = distension
‘) N = number
-...-~-..- .. .. .--_ ,.-_ ._..__. -.-- .I..._..._- ..__
.._.
0.00500 0.00250
1 6 16
0.00100 0.00075 o.clooso -_-I_---_-
_.-______----
-76.1126 -76.1114 -76.1113
-76.1488 -76.1306 -76.1208
0.0374 0.0192 0.0094
52
-76.1113
-76.1!35
0.0024
72
-76.11.14
-76.1119
@.0005
97 -76.1114 -76.1114 .--.-.-._.-._ -._._- ._,,.___._
0.0
_
for I-I,0 (IV = 371)
particular, the second mcst do~~inant term in the 03 wavefurxtion is a double excitation with mixing coefficient of 0.33, and comprises tk second configuration for the K‘= 2 zero order prbblem. giv&n in table 1. An inherent pzrt of the perturbation approach is the choice of the zero order problem. In fact, this .choice could be made in several way$ based on the en.’ is 0.959 1; In
..’
_.
.
era expansion. i: presc~p~ion svhich is both strai&tforward and which seems to lead to desirabIe.convergence properties for the energy expansion was defined and applied iii the course of our czlctilations. A zero order set contbing only singles and (oubies.was deter-. knined by seiecting those configurations -who% secdn,d
(
‘..
Volume 37, number
1
order energy contributions exceeded a threshold tI *. this number by K, . Note that this requires of only the first row and the diagonal of the singles and doubles ff-matrix. The second order contri-
We denote knowledge
butions for each configuration in the full list was then obtained relative to this zero order problem, and a new choice of the zero order problem was made, which consisted of a!1 those configurations whose energy contributions exceeded a second threshold, r,_ This second step is necessary to determine which triples and quadruples should be included in zero order, and it requires a knowledge of only the fist K1 columns as well as the diagonal of the H-matrix. Lastly, using the total R-configuration zero order problem, the full perturbation calculation was performed. Tie results of table 2 indicate the nature of the ccnvergence of the perturbation series for the NO5 and 0, cases, when the zero order problem is chosen according to the above prescription. For these calculations, the thresholds used were f1 = 0.01 au and l2 =O.OOl au. It is seen that, in at least these test cases, the cumulative contribution to the energy by terms beyond second order is small. This suggests that truncation of the perturbation sum after second order could yield a good apprcximation to the actual value of the root. In fact, for NO2 and O,, the sums through second order are -203.7835 au and -224.4561 au respectively, both well within a millihartree of the actual values.
In table 3, the dependence of the second order results on the value of K, hence I 1, is illustrated using the 371 term CI calculations on H1O. Since these calculations involve only single and double excitations from the SCF determinant, consideration of the threshold, f7, is not necessary. From this table, it is clear that while the problem is well conditioned with respect to the full fifth order results, the second order truncation shows a strong dependence on K. This illustrates the fact that if one carries out second order calculations only, the results will depend on the values taken for tl and t2. Moreover, the appropriate values will vary from molecule to molecule. In actual calculations one could proceed in two ways: either choose tl and t2 to be sufficiently small so that K includes all important configu-
rations to begin with; or (a) choose K, to correspond to those singles and doubles which account for about * Note that negative.
all the second
I January
CHG~IlCAL PHYSICS LETTERS
order energ
contributions
must be
1976
l/4 of the second order energy with respect to K = 1, and then (b) choose c2 by requiring the zero order problem to be constructed from those configurations which account for a large fraction (we ha& found 3/4 to be sufficient) of the second order energy with respect to the zero order problem chosen in (a)_ Another (insurance) step can be added by screening the second order energies obtained with this first guess of CI and t2 and increasing the value of K, if necessary. Except for the manner in which K is chosen, this second order truncation procedure is equivalent to the AK method of ref. [3] *-. Th.e significance of the above results is apparent when one considers the reduced effort which the second order calculations imply. Denoting the number of single and double excitations in the full list by N,, the number of matrix elements needed in the three steps listed above are WI, N(K1 + 1) - Kl(Kl + 1)/Z, and IV(K + 1j - K(K+-1)/2, respectively. This assumes that no elements are saved from step to step. The sum of these numbers is to be compared with P4)V + 1)/2 matrix elements needed for the full calculation. For the NO; and 03 test cases, the comparisons are respectively: 10707 versus 25200 matrix elements and 7194 versus 19701 matrix elements. These differences are even more dramatic if the full CI list does not include trip!es and quadruples. Another point which we have observed is that if the zero order problem is chosen as discussed above, quite accurate values for the root can be obtained if the energy expansion is truncated at fourth order. Unlike the second order truncation, this requires that the entire H-matrix be available; however, the fourth order calculation involves only a single pass over H, while the lift!? requires two.
For those applications in which the full fifth order calculation is necessary, the amount of computer time required as compared to that needed for actual diagonalization is of interest. In our work, the H?O calculation (with A’= 3238 and K = 1) provides the only real basis of comparison, as that is the only case in which the requisite computer programs invo!ved similar levels of optimization. in this czye, we found that the fifth order ca!culation required about I rninutc of combined CPU/IO
time, while the time required
for the actual
** We arc indebted to Professor i. Shavitt for this equivalence [$ I.
pointing
out
113
Volume 37, number
1
CHEhWAL
diagonalizarion tias about 3 minutes?. Both programs required about the same amOunt of core.-It should be noted, however, that the reletive time requirements will v&y with the values of K and N. The first application of a method such as ours involvtid the second order calculations of Boys ft;] . Our formulation differs from his both because we include hio&er order terms and because we explitiitly extend ihe development to K > 1. Other methods of estimating energy contributions require prior diagonalization of the N-matrix, or some (still Iarge; but sparse) truncation of it [3j _ in any case, tile sum of the energy contributions does not equal the actual root. Tllis latter restriction is removed in the method of Bunge [7] but his method also requires knowledge of the eigenvectors and can (and often does) result in positive contributions from some of the configurations: It is however assumed that such configurations contribute very little to the Ci energy lowering and can be deleted. Positive energy contributions can occpr in our method also;however if the zero order problem is chosen correctly, the positive contributions (through Fifth order) are on the order of I Od6 au. Finally, while this work was in progress, a method equivalent to third order Br~lIou~n-W~gRer perturbation theory was pubIished by Segal and Wetmore [8j . .Their method, which is iterative, leads to somewhat lesser accuracy. relative to ours for a given value of K.
The only specific system common to both their work and ours involves the X1 .A1 state of H20. In their case, using a 40 term zero order
hmction
(out of429
the 0.0012 au accuracy for Cc= 1 ar,d I\‘= 371 and the 0.0035 Z.IIJaccuracy for K = 1 and N = 3235 in our calcollations [see tables 1 and 3‘). Both calculations included only single and double excit:ltions from the clcsed ~1x11 SCF .determinanr. The dossibility of effective trunca-
+ both the dingon;llharion
(convergence to 10°3 in the eiqenvectors) and thr: energy contrillution calculations were wrricd out on an IRhl 370/195 cmiputer, using programs compiled under f-‘ORTRAN-ii (OPT = 2). The dineonnlization prosram used \y.vaiHCEN written by Shavitt [Sl
.I
Ii4
..‘_
_,
: :
.’
,.
::
,‘. .,’ .,”
‘,
.’
.’ _.. .‘
”
:
_,: ..
tion of their perturbario~ demonstrated.
sum at second
1976
order was &so
4. Conclusions In this wcrk we have presented a perturbative alternative to diagonalization of’ a CI matrix. This method can be applied to the selection of configurations prior
to diagonalizaticrn with concomitant reduction in the computational effort, as well hs ftdfiihg an interpretive function. We have also set forth a method for defining an effective zero order problem so that depending on the precision desired in a particular application or that warranted by the rest of the calculation, truncation at second order can replace diagonalization of the CI matrix using only a fraction of tile total nuzzi5zi of matrix elements. Further, for those cases in which greater accuracy is desired, we have demonstrated that fourth order theory, requiring only a single pass Jver the firmatrix can be sufficient. At the present time, we are continuing tests of this method, and extending calculations to include perturbat&e compLltation of the Cl mixing coefficients through second order, both for use as starting guesses
in the standard diagonalization procedure, and as a source of approximate natural orbit&.
References
total
con.figurations) t!ley obtained results within 0.0020 au of the diagonalized value..This can be compared with
..
I Jttnuary
PHYSICS LETTERS
,,
[l ] 1.0. liirschfclder, W. Byers-Brown and S.T. Epstein, Advan. Quantum Chcm. 1 (1964) 155. [2/ I. Shsvitt, CF. Bwder, A. Pipano and R.P. Hosteny, J. Comp. Phys. 11 (1973) PO. [ 3! 2. Gershsorn and I. Shavitt, Intern. f. QL~n~um Chem. 2 (1968) 751; R.E. Brown. Ph.D. Thesis, Indinna Univcrsiry (1967). [4] 1-R. Kahn, P.J. Hay and I. Shavitr, J. Chem. Phys. 61 (197513530. [S] 1. Shavitr, EIGEN, QCPE Program -’ 171, Cjuantum Chemistry Program Eschange,‘lndianz University, Bloomingion, Indiana. [6] S.F. Boys, Proc. Roy. Sac. AZ17 (1953) 136. [7J C’.t”. Bunge, Phys. Rev. 168 (1968) 92. (81 G.A. Se@ and R.W. Wetmore, Chcm. Phys. Letters 37 (1975) 556.
1
1 January
CHEMICAL PHYS!CS LETTERS
1976
:
APPLICATION OF PELZTURBATION THEORY IN LARGE CONFIGURATION !NTERACTION CALCULATIONS+ PJ.
FORTUNE
and Bruce J. ROSENBE.RG
Chemistry Division, Argotwe
National Laboratory.
Argome.
Illinois
60439.
USA
Received 2s JUy 1975 Revised manuscript received 20 August 1975
Ray!aigh-Schrijdinger Perturbation theory has been applied through fifth order in the energy, to the problem of estimating the rodts of the secular equation in large configuration interaction calculations. The NO:, 03 and Hz0 molecules are used as test uses, wit% accuracy as good as O.Gl eV, with appropriate choice of zero order problem.
1. Introduction The principal technicye used to include electron correlation effects in theoretical molecular calculations is
test calculations. The results of the test calculations performed in this work further indicate that, rvith an appropriate defini-
the configuration interaction (C!) form of the linear variational me*Lhod. The number of configurations in a given czilcula t-IOII depends on t,‘le level of accuracy with which one is concerned. It is not uncommon for
tion of the zero order problem, second order perturbation theory can be sufficient for some applications. For such cases, accuracy on the order of a millihartree can be achieved while constructing only part of the full Hmatrix.
this number to reach several thousand, so that application of configuration selection techniques to minimize the size of the secular determinant prior to diqonaliza-
2. Theory
tion can be important. Similarly, it would be desirable to be able to obtain reliable estimates of the roots of a large CI matrix without using (or therefore construct-
ing) the entire matrk. In this work we have appliad
Rayleigh-Schredinger perturbation theory (RSFT) [l] through fifth order to the matrix eigenvalue problem and expressed the resulting total energy in terms of a sum of en&y contributions from individual confi;urations. Tnese energy contributions can be used to eliininate a subset of con-
figurations from the original
CI list. For large matrices,
the method
requires a fraction of the time needed for diagonalization via the usual iechtiques [2] , with the difference between the perturbation results and those from the full diagonalization being quite small in our
The application of RSPT to the problem of determining energy contributions of individual configurations is natural. The problem at hand is one of solving the secular equation: Hc = EC.
(1)
To
apply RSPT, we separate the H-matrix into a sum of zero order plus pertllrbing terms: H= Ho +AV.
(3
Here, X is a formal ordering parameter which, in the fki analysis, will be set to one. With the diiension of the H-matrix denoted by N, we take the form of HO and
v to be:.
* Based on work performed unde; the auspices of the Uaited ,Setes Energy Resev&.and Davelopmen! Administration. .. .
.llO’.,._ ,:,-_
: -. ‘. _._ ..-:..y
‘.:. .,: .,:_.-
-y.,
_’
:
y,. ..:
,, .,
,.
.-.,
.
.;
1
;
: _..’
Volume
cmwa_
37, number 1
HI7 ...HIK 4, ._ !G __.H,K Hll
_d
.
.
.
.
0
_.. 0
0
. .. 0
._
..-;
(HK1HK2...~;
Ho=
,jQ:
0;
:
yY’Tl,K+l,‘-‘;
PHYsIcs LETTERS
I January 1976
1
(3)
i
,
and V=H-HO. The zero
(4)
order
problem
is then
one of diagonahzing
the (relatively small) K X K submatrix of H, with Jth eigenvalue denoted by ei”), and eigenvectors represented by:
Note that in the expression for @J, the last two bracketed terms give rise to equal sums; however, for the purof defining energy contributions, bine them. Defming
pose
we do not com-
00)
(5)
we write the energy contribution S,, of the r,th configuration, order by order, as follows:
and c$o’ = [0 . ..O....
l,...OIT
J>K.
(6)
In what follows, we assume that the configuration list is ordered so that the first K terms correspond to the dominant set. Ways in which this set can be chosen will be discussed in a later section of this paper. For the sake of simplicity, be the lowest
we take
the root
of interest
to
one.
With the definition that the derivative of a vector is the vector of the derivatives of the individual components, one easily- obtains the usual perturbation equations [ I] _One can expand the solutions to these equations in terms of the zero order vectors, and defining (7) and
it can be shown that the expressions for the first five corrections to the lowest root are [l] :
(11) The-total lowest
contribution root
s,r= i:
is then
of the rzth configuration defined
to the
by:
s,‘:“.
111= I Using eqs. (3) and C4), it is straightfo~vard the various matrix (11) are given by: ‘Jr,=
’
products
to show
that
P’,, which enter eqs. (7)-
(i3n) R Cl3b)
=HJIC1 -S/r)
3. Calculation
J>K.I>K.
(13c)
and discussion
For the purposes
of testing
the efficacy
of the above
i
-: Vulumc 37. notiber :
CIIEWCAL
i Jenwry
PHYSICS LETTERS
1976
Tab!.e 1
.. Results of perrurbation
calculations
State
Molecule
___._...-,_
_
eqs. (7)-(13)
using IF)
.-..
.._..
.-.... .
Nb) Epert .. . . .-.- .- --. . ..- ..-..I.---
lA&Vlid~
‘diag
. NO;
X!Al
1
324
-203.6683
-103.7829
3.111
NO;
X’A,
-203.7825 -224.6105
-224.4363
O.Oi 1 4.194
9s .HzO
X’A, X’AI X’AI X’AI
224 198
-203.7829
03
33 1 2 24 1
198 198 371
-224.4463 -224.4555’ -76.1126
-224.4563 -224:456;3 -76.1114
0.272 0.022 0.033
H,O -~--
X’A, -... -__-.._-._
1
323sc)
-76.3009
-76.2974
0.095
03.
of
cf
contigusatiols.
Calculation includes singe and double excitations d, 1 eV= 23.2558 kcal/mole.
Table 2 Order-by-order
perturbation
enxgies
03
--.-----_-...--
only. Present program limitations
estension
of this we
--
eS3)
$)
$)
-203.76597
-0.01753
0.00226
-0.00219
b.00094
-224.43616 .---..-..--.
-0.01997
0.00204
-0.00x7
0.00111
-._...- ---------_.-._--__.
.--...--. ---
H20, NO; and O3 molecules are summarized in tabie 1. For the NO; and 03 calculations, the convergence
of the perturbation
expansion is ~Iustrated in table 2. 3 lists the dependence of the perturbation results on K for the H,O calcu!ations. The results listed in table 1 indicate that the NO; and 0; problems with K = 1 are the most poorly conditioned cases of those which we considered. This is also manifested by the fact that the squares of the dom-
.__.____-____-__________
Table 3 Perturbation results as a function of K for the Iv = 371 [iz(.j CI calculations”) _-_.._..., ----__-_------
---.---7-s---.
Tabk
inant C’Imiving coefficient respectively,
for these cases are 0.8579 while
that
to K > 1.
---___
$)
method, we have applied it to several cases of chemical interest. These calculations, on the X1 Al states of the
and 0.8225,
prohibit
for NOf (K = 33) and 03 (K = 24) caIc&tions
$)
+ N32
-----
of zero order problem.
‘)K. = distension
‘) N = number
-...-~-..- .. .. .--_ ,.-_ ._..__. -.-- .I..._..._- ..__
.._.
0.00500 0.00250
1 6 16
0.00100 0.00075 o.clooso -_-I_---_-
_.-______----
-76.1126 -76.1114 -76.1113
-76.1488 -76.1306 -76.1208
0.0374 0.0192 0.0094
52
-76.1113
-76.1!35
0.0024
72
-76.11.14
-76.1119
@.0005
97 -76.1114 -76.1114 .--.-.-._.-._ -._._- ._,,.___._
0.0
_
for I-I,0 (IV = 371)
particular, the second mcst do~~inant term in the 03 wavefurxtion is a double excitation with mixing coefficient of 0.33, and comprises tk second configuration for the K‘= 2 zero order prbblem. giv&n in table 1. An inherent pzrt of the perturbation approach is the choice of the zero order problem. In fact, this .choice could be made in several way$ based on the en.’ is 0.959 1; In
..’
_.
.
era expansion. i: presc~p~ion svhich is both strai&tforward and which seems to lead to desirabIe.convergence properties for the energy expansion was defined and applied iii the course of our czlctilations. A zero order set contbing only singles and (oubies.was deter-. knined by seiecting those configurations -who% secdn,d
(
‘..
Volume 37, number
1
order energy contributions exceeded a threshold tI *. this number by K, . Note that this requires of only the first row and the diagonal of the singles and doubles ff-matrix. The second order contri-
We denote knowledge
butions for each configuration in the full list was then obtained relative to this zero order problem, and a new choice of the zero order problem was made, which consisted of a!1 those configurations whose energy contributions exceeded a second threshold, r,_ This second step is necessary to determine which triples and quadruples should be included in zero order, and it requires a knowledge of only the fist K1 columns as well as the diagonal of the H-matrix. Lastly, using the total R-configuration zero order problem, the full perturbation calculation was performed. Tie results of table 2 indicate the nature of the ccnvergence of the perturbation series for the NO5 and 0, cases, when the zero order problem is chosen according to the above prescription. For these calculations, the thresholds used were f1 = 0.01 au and l2 =O.OOl au. It is seen that, in at least these test cases, the cumulative contribution to the energy by terms beyond second order is small. This suggests that truncation of the perturbation sum after second order could yield a good apprcximation to the actual value of the root. In fact, for NO2 and O,, the sums through second order are -203.7835 au and -224.4561 au respectively, both well within a millihartree of the actual values.
In table 3, the dependence of the second order results on the value of K, hence I 1, is illustrated using the 371 term CI calculations on H1O. Since these calculations involve only single and double excitations from the SCF determinant, consideration of the threshold, f7, is not necessary. From this table, it is clear that while the problem is well conditioned with respect to the full fifth order results, the second order truncation shows a strong dependence on K. This illustrates the fact that if one carries out second order calculations only, the results will depend on the values taken for tl and t2. Moreover, the appropriate values will vary from molecule to molecule. In actual calculations one could proceed in two ways: either choose tl and t2 to be sufficiently small so that K includes all important configu-
rations to begin with; or (a) choose K, to correspond to those singles and doubles which account for about * Note that negative.
all the second
I January
CHG~IlCAL PHYSICS LETTERS
order energ
contributions
must be
1976
l/4 of the second order energy with respect to K = 1, and then (b) choose c2 by requiring the zero order problem to be constructed from those configurations which account for a large fraction (we ha& found 3/4 to be sufficient) of the second order energy with respect to the zero order problem chosen in (a)_ Another (insurance) step can be added by screening the second order energies obtained with this first guess of CI and t2 and increasing the value of K, if necessary. Except for the manner in which K is chosen, this second order truncation procedure is equivalent to the AK method of ref. [3] *-. Th.e significance of the above results is apparent when one considers the reduced effort which the second order calculations imply. Denoting the number of single and double excitations in the full list by N,, the number of matrix elements needed in the three steps listed above are WI, N(K1 + 1) - Kl(Kl + 1)/Z, and IV(K + 1j - K(K+-1)/2, respectively. This assumes that no elements are saved from step to step. The sum of these numbers is to be compared with P4)V + 1)/2 matrix elements needed for the full calculation. For the NO; and 03 test cases, the comparisons are respectively: 10707 versus 25200 matrix elements and 7194 versus 19701 matrix elements. These differences are even more dramatic if the full CI list does not include trip!es and quadruples. Another point which we have observed is that if the zero order problem is chosen as discussed above, quite accurate values for the root can be obtained if the energy expansion is truncated at fourth order. Unlike the second order truncation, this requires that the entire H-matrix be available; however, the fourth order calculation involves only a single pass over H, while the lift!? requires two.
For those applications in which the full fifth order calculation is necessary, the amount of computer time required as compared to that needed for actual diagonalization is of interest. In our work, the H?O calculation (with A’= 3238 and K = 1) provides the only real basis of comparison, as that is the only case in which the requisite computer programs invo!ved similar levels of optimization. in this czye, we found that the fifth order ca!culation required about I rninutc of combined CPU/IO
time, while the time required
for the actual
** We arc indebted to Professor i. Shavitt for this equivalence [$ I.
pointing
out
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CHEhWAL
diagonalizarion tias about 3 minutes?. Both programs required about the same amOunt of core.-It should be noted, however, that the reletive time requirements will v&y with the values of K and N. The first application of a method such as ours involvtid the second order calculations of Boys ft;] . Our formulation differs from his both because we include hio&er order terms and because we explitiitly extend ihe development to K > 1. Other methods of estimating energy contributions require prior diagonalization of the N-matrix, or some (still Iarge; but sparse) truncation of it [3j _ in any case, tile sum of the energy contributions does not equal the actual root. Tllis latter restriction is removed in the method of Bunge [7] but his method also requires knowledge of the eigenvectors and can (and often does) result in positive contributions from some of the configurations: It is however assumed that such configurations contribute very little to the Ci energy lowering and can be deleted. Positive energy contributions can occpr in our method also;however if the zero order problem is chosen correctly, the positive contributions (through Fifth order) are on the order of I Od6 au. Finally, while this work was in progress, a method equivalent to third order Br~lIou~n-W~gRer perturbation theory was pubIished by Segal and Wetmore [8j . .Their method, which is iterative, leads to somewhat lesser accuracy. relative to ours for a given value of K.
The only specific system common to both their work and ours involves the X1 .A1 state of H20. In their case, using a 40 term zero order
hmction
(out of429
the 0.0012 au accuracy for Cc= 1 ar,d I\‘= 371 and the 0.0035 Z.IIJaccuracy for K = 1 and N = 3235 in our calcollations [see tables 1 and 3‘). Both calculations included only single and double excit:ltions from the clcsed ~1x11 SCF .determinanr. The dossibility of effective trunca-
+ both the dingon;llharion
(convergence to 10°3 in the eiqenvectors) and thr: energy contrillution calculations were wrricd out on an IRhl 370/195 cmiputer, using programs compiled under f-‘ORTRAN-ii (OPT = 2). The dineonnlization prosram used \y.vaiHCEN written by Shavitt [Sl
.I
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_,
: :
.’
,.
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tion of their perturbario~ demonstrated.
sum at second
1976
order was &so
4. Conclusions In this wcrk we have presented a perturbative alternative to diagonalization of’ a CI matrix. This method can be applied to the selection of configurations prior
to diagonalizaticrn with concomitant reduction in the computational effort, as well hs ftdfiihg an interpretive function. We have also set forth a method for defining an effective zero order problem so that depending on the precision desired in a particular application or that warranted by the rest of the calculation, truncation at second order can replace diagonalization of the CI matrix using only a fraction of tile total nuzzi5zi of matrix elements. Further, for those cases in which greater accuracy is desired, we have demonstrated that fourth order theory, requiring only a single pass Jver the firmatrix can be sufficient. At the present time, we are continuing tests of this method, and extending calculations to include perturbat&e compLltation of the Cl mixing coefficients through second order, both for use as starting guesses
in the standard diagonalization procedure, and as a source of approximate natural orbit&.
References
total
con.figurations) t!ley obtained results within 0.0020 au of the diagonalized value..This can be compared with
..
I Jttnuary
PHYSICS LETTERS
,,
[l ] 1.0. liirschfclder, W. Byers-Brown and S.T. Epstein, Advan. Quantum Chcm. 1 (1964) 155. [2/ I. Shsvitt, CF. Bwder, A. Pipano and R.P. Hosteny, J. Comp. Phys. 11 (1973) PO. [ 3! 2. Gershsorn and I. Shavitt, Intern. f. QL~n~um Chem. 2 (1968) 751; R.E. Brown. Ph.D. Thesis, Indinna Univcrsiry (1967). [4] 1-R. Kahn, P.J. Hay and I. Shavitr, J. Chem. Phys. 61 (197513530. [S] 1. Shavitr, EIGEN, QCPE Program -’ 171, Cjuantum Chemistry Program Eschange,‘lndianz University, Bloomingion, Indiana. [6] S.F. Boys, Proc. Roy. Sac. AZ17 (1953) 136. [7J C’.t”. Bunge, Phys. Rev. 168 (1968) 92. (81 G.A. Se@ and R.W. Wetmore, Chcm. Phys. Letters 37 (1975) 556.