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Accelerating the convergence of matrix Hartree-Fock calculations
CHEMICAL PHYSICSLETTERS
Volume 8. number 2
ACCELERATING MATRIX
THE
HARTREE-FOCK
15
CONVERGENCE
January 1971
OF
CALCULATIONS
*
N. W. WINTER and T. H. DUNNING Jr. BattelIe
Memorial
Institute,
Columbus,
Ohio 43201,
UsA
Received 1 December 1970
An extrapolation method based on the ek-transformation 1s applied to matrix Hartree-Fock calculations on the water and formaldehyde molecules. For water the two-electron energy is presented at each iteration for calculations with and without extrapolation. Results are also given to demonstrate the effectiveness of the method for different choices of starting vectors. Experience has shown that each extrapoIation usually saves from focr to seven iterations.
1. INTRODUCTION Implicit in any self-consistent-field (SCF) caIculation is an iterative procedure requiring a guess for the occupied orbitals which is then used to calculate a potential function from which we obtain an improved set of orbitals. With the increased efficiency of molecular integral programs, this iterative process represents a significant portion of the overall calculation time. For many systems of chemical interest, even with moderate coxtraction [l] and a good initial guess, it is not unusual for the iterative solution of the. matrix equations to take twice as long as the calculation of theintegrals. Therefore, it is important that an effort be made to reduce the SCF part of the calculation. In order to accelerate the convergence of matrix Hsrtree-Fack calculations, several approaches have been taken: quadratically convergent SCF schemes [Z], improved initial guesses for the vectors, and extrapolation techniques [3]. The use of good initial vectors is the most obvious way to reduce the number of iterations, and in addition, reduces the possibility of trapping of the solution in a local minimum. Quadratically convergent methods aIs0 substantiaIly reduce the number of.iterations required for convergence. However, the time per iteration is signjficantly greater. Extrapolation is widely used to predict an improved set of vectors using the information * Part -of this work was c&ied out while the authors were at the California &&itutc of Technoloti, Pasadena, Callfornla. USA.
from previous iterations, but not all extrapolation methods are as reliable as one would like [4].
The method discussed
here has shown both its
usefulness in predicting an improved set of coefficient vectors and its reliability in numerous molecular calculations.
2. DESCRIPTION
AND APPLICATION
OF THE
METHOD As previously
mentioned, the use of good can result in a considerable reduction in the number of SCF iterations. We have chosen the equivalent orbitak (EO’s) proposed by Letcher and Dunning [5] as the initial guess for starting vectors
the water calculation. The charge distribution predicted by the EO wavefunction is compared to that obtained from the Hartree-Fack calculation
in table 1. The energy is within U-O.2 au of the converged value and the agreement between the one-electron properties is reasonable. Although the EO wavefunction cannot be considered more than a fair representation of tie molecular wavefunction, it is clearly an adequate initial guess. Cf course, there are other means of generating satififaCtOrY starting vectors, such as extended Hiickel[6], Wolfsburg-Relmhoftz [7], and the NEMO [8] methods. The important point is that any guess which gives a reasonabIe charge distribution is decidedly superior to the use of zero vectors. This will be quantitatively illustrated in the last section after the discussion of the extrapolation method. 169
Yolunie 8,.number‘2 ‘-
;_.
-_
CHEMICALPHYSICSLETTERS
15Januaty1971
Table1
Comparison of the expectation valuei of varlouaopera.tors for the equivalent drbitalandHartree-Fockwavefunctionsof water
-
Expectationvaluea*b) Operator %iF
-%-0 -75.8297'7
LCAOMO -76.00696
cgl
-V/T
2.00202
2.00003
=0 2 *cm
1.4307
1.1744
8,0983
7.2141
5.6817
5.4599
6.8151
6.5122
3Y,2m -em
-1.7750
-1.3132
32zm - +gm
-0.0742
-0.1552
W-0
23.4138
23.4395
l&H
5.5607
s2 cm 2 +rn
-
cc?+1 + z-l
1 cm+1 _Cnt ’ 2 t
(2)
where C?I is taken as zero. The transformation array can be written in the form of an extrapolation table as shown below for five iterations, 4 CY c;
c;
(3)
5.72'70
0.3626
-0.2353
%H/+i
1.8906
1.9576
IH/4
1.4142
1.4595
@q-_r2H)/$
-1.0613
-1.0959
Wj!p-~)/~~
2.1073
2.1'769
9
1056'
2QiSl
W~-r20)/r&
-2.6462
-1.6117
(3+j -Yg,/Yh
-
0.3157
-0.1545
W-H)
0.2140
0.3430
W-0)
292.5101
294.1794
Q90
represent a particular .molecular orbital coefficient as a function of iteration. We then define tile elements of the transfbrniation ‘array as foliows:
a) Only the electroniccontributiona have been given.
b) All resultsare In atomic units.
T&e extrapolation procedure has been dealt with in detail by Shanks [9] and by Wynn [lOJ in connection with divergent and slowly convergent sequences. Petersson and McKay [ll) have also applied.the method to the calculation of molecular integrals. It is‘a non-linear sequence-tosequence transformation denoted as..the eptrans: formation by Shanks. In application to iterative processes, this method is also referred tb as : Aitken’s gencr&ed 6.2process; .‘: .: In order to describe the_ method let the se: ,--_ :. .
The even columns are new sequences of values, each of which converges more rapidly than the previous. Each sequence is transformed according to eq. (2) to produce the following cohnnn. This algorithm for evaluating the transform is due to Wynn [lo]. The element Co is taken as the extrapolated coefficient and save % for use in the next SCF iteration. Because of the oscillatory nature of the molecnlar orbital coefficients as a function of iteration, we have found the B-term transformation to be the most useful in matrix Hartree-Fock calculations. The ek-transformation has been incorporated in our matrix Hartree-Fock routines for a number of years. During this time it has been used in calculations on numerous closed shell molecules ranging in size from first-row Piatpmics to butadiene. In each case the pattern of convergence and the effectiveness of the method were essentially the same. :For this reason we report detailed. information on only one example. UsinglIuzinagals [lZ] .(9s5p)-basis set Ion oxygen and a 3s. basis. scaled to:l.O on ,the hydrogens, a complete.SCF calculation was carri,ed out dn H30. bothwith and_witbo_utktrapolation. : In add$ion, two different.sets’ of starting vectors . \yere.‘y&d f or.,ea;ch. calcGla.i~oii; : Since *e saving8 i- afforded by;_t&eextrapolation depends on the
Volume 8, number 2
The two-electron Iteration 1
2 3 4
5
6 7 8 9 10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25
CBEMICAL PHYSICS LETTERS
15 January1971
Table 2 energy at each iteration for the water ca.IcdatIonsaP)
Case A 69.8998 21.3946 52.9234 26.6997 47.0959 31.6084 42.0668 35.0469 39.5305 36.6700 38.4716 37.3369 38.0496
37.6019 37.8830 37.7065 37.8173 37.7477 37.7914 37.7640 37.7812 37.7704 37.7772 37.7729 37.7756
Case B
37.8441 37.7342 37.8000 37.7505 37.7715 37.7763 37.7735
Case C 37.6138 37.8277 37.7279 37.7994 37.7573 37.7847 37.7679 37.7786 37.7720 37.7762 37.7736 37.7752 37.7742 37.7748 37.7744
Case D
37.7768 37.7735 37.7753 37.7741 37.7748 3717745 37.7746
4
6
s
10
12
14
16
ITERATION
Fig. 1. The total energy of the fcrmaldehyde molecule.
a) Case A: zero vectors and no extrapolations; case B: zero vectors followed by extrapolation of iterations 6-16 and 10-14: case C: EO vectors and no extrapolations; case D: EO vectors followed by extrapolation of iterations 2-6 and 6-10. b) The results are given in atomic units. initial guess, the calculations were started zero vectors as well as the EO’s.
”
with
In order to display the convergence characteristics of the electronic charge distribution, table 2 gives the two-electron energy at each iteration. This quantity is far more sensitive to changes in the wavefunction than is the total energy. The first column lists the two-electron energy for the calculation starting with zero vectors and with no extrapolations. After 25 iterations the total energy has converged to 3 X 10m8 but the two-electron energy has not yet converged past the third decimal place. In the second column the calculation is repeated except that iterations 6-10 and then lo-!4 were transformed according to eq. (2). After the first extrapolation the two-electron, energy is as well converged as that of iteiation 17 in the first column. The two iterations after the second extrapolation are as well converged as iterations 24-25 of the first case; Overall the ko extrapolations .saved about nineriterations..
The third column contains the results for the calculation starting with EO vectors and without extrapolation. After 15 iterations t$z total energy has converged to better than LO . The use of good initial vectors saved more than ten iterations relative to starting with zero vectors. The fourth column gives the results for th? same calculation except that a 5-term transformation was carried out on iterations 2-6 and on iterations 6-10. The first extrapolation saved approximately four iterations and the second extrapolation then converged the two-electron energy to the number of figures quoted. Using one extrapolation and three additional iterations, the energy converged to eight decimal places. This rate of convergence is typical of the calcuiations on other molecules that we have .studied, Le., 8-10 iterations inture convergence of the total energy to about 10’ . In an attempt to apply the method to a divergent case, a calculation on the formaldehyde molecule in a contracted [3s2p/2sl basis was carried out starting with zero vectors:. Such an initial guess appears to be partictiarly inappropriate for formaldehyde. As can be seen in fig. 1, the energy began to diverge, oscillating about a line .with a divergent slope for the first twelve iterations. The vectors from iterations 8-12 were subjected to a 5-term transformation‘and the calculation was conttiued. The energy plotted at iteration 12 was-found using the results of the extrapolation mid represents the antiIimit of the diverging- sequence. .Ttie remaining iterations converge to an energy. of -,144.86495 au. Besides demonstrating a unique feature of the extrapola-
_, ,Xqh+pe 8,‘neber ;. --,_. __-
Z-
._
:, I’
CHErI~A~-P~~IC~-.~E~E~
.-. ._ “:*-tion x&hod, this e&3&e a&o +ws the impor: tancie .of .a ‘good initial guess’ for ,the occupied or.:Tbjtals. . ” : ;.‘.
:
-.-
.~ .._
AC~CNOWLEDGEMEN~
1'
The tiuthors~gratefully aknowledge many. . helpful discuss~qns with fi. George Petersson and Dr. Vintient. McKay. Fitiancial support from department of the California Institute af Technology is &o acknowledged.
_ the -Fhex&stj
REFERENCES.
_
a
[1] ‘X.3. DwuttugJr., J. Chem. Phys. 53 (lS70) 2823.
, .
1
‘_
:-
1
-:
‘15 January 1971
[ii W,R..&&&l, J; Chem. Pi&& 47 (lQ&).32&3. [3] C, C;~;Roothaan and P.S. Bagus, in: Methods in comp&affom&l physfca, Vol. 2, eda. B;Alder, S. Ferabach and M. Rate&erg (Academfa Prass. New : I Yoxk, 1993) p. 47; _[4] yix3ildor ~d_~_~h~Vitt, J* Cbem. Pbys. 44 &66)
j5J J. H:Lktche~ and i. H. DuqningJr., J. C&n. Phye.
, 48 (1968) $538. [6] R.Iiaffman. J. Cbem. Phys. 39 (1964) 1397; 40 (1964) 3247; 2474, 2480. j?] M. WoIfsberg and L.HeImholz, J. Cbem. Pbye. 20 (1952) 837.. [&jM. D. Newt&, F. P. Boer and W. N. Lipscomb, J, Am. Cbem. Sot. 88 (1966).2253, 2361, 2367. [S] D.Sban&t, a.-M&b. Phys. 34 (1955) 1. [lo] P. Wm. Math. Ak?s Computatfon 64 (1956) 91. [ll] G. A. Petersson and V.McKoy, J. Chem. Phys. 46
(1967) 4362.
[li] S.FIuzlna&a,J. Chem. Phys. 42 (l965) 1293,
Volume 8. number 2
ACCELERATING MATRIX
THE
HARTREE-FOCK
15
CONVERGENCE
January 1971
OF
CALCULATIONS
*
N. W. WINTER and T. H. DUNNING Jr. BattelIe
Memorial
Institute,
Columbus,
Ohio 43201,
UsA
Received 1 December 1970
An extrapolation method based on the ek-transformation 1s applied to matrix Hartree-Fock calculations on the water and formaldehyde molecules. For water the two-electron energy is presented at each iteration for calculations with and without extrapolation. Results are also given to demonstrate the effectiveness of the method for different choices of starting vectors. Experience has shown that each extrapoIation usually saves from focr to seven iterations.
1. INTRODUCTION Implicit in any self-consistent-field (SCF) caIculation is an iterative procedure requiring a guess for the occupied orbitals which is then used to calculate a potential function from which we obtain an improved set of orbitals. With the increased efficiency of molecular integral programs, this iterative process represents a significant portion of the overall calculation time. For many systems of chemical interest, even with moderate coxtraction [l] and a good initial guess, it is not unusual for the iterative solution of the. matrix equations to take twice as long as the calculation of theintegrals. Therefore, it is important that an effort be made to reduce the SCF part of the calculation. In order to accelerate the convergence of matrix Hsrtree-Fack calculations, several approaches have been taken: quadratically convergent SCF schemes [Z], improved initial guesses for the vectors, and extrapolation techniques [3]. The use of good initial vectors is the most obvious way to reduce the number of iterations, and in addition, reduces the possibility of trapping of the solution in a local minimum. Quadratically convergent methods aIs0 substantiaIly reduce the number of.iterations required for convergence. However, the time per iteration is signjficantly greater. Extrapolation is widely used to predict an improved set of vectors using the information * Part -of this work was c&ied out while the authors were at the California &&itutc of Technoloti, Pasadena, Callfornla. USA.
from previous iterations, but not all extrapolation methods are as reliable as one would like [4].
The method discussed
here has shown both its
usefulness in predicting an improved set of coefficient vectors and its reliability in numerous molecular calculations.
2. DESCRIPTION
AND APPLICATION
OF THE
METHOD As previously
mentioned, the use of good can result in a considerable reduction in the number of SCF iterations. We have chosen the equivalent orbitak (EO’s) proposed by Letcher and Dunning [5] as the initial guess for starting vectors
the water calculation. The charge distribution predicted by the EO wavefunction is compared to that obtained from the Hartree-Fack calculation
in table 1. The energy is within U-O.2 au of the converged value and the agreement between the one-electron properties is reasonable. Although the EO wavefunction cannot be considered more than a fair representation of tie molecular wavefunction, it is clearly an adequate initial guess. Cf course, there are other means of generating satififaCtOrY starting vectors, such as extended Hiickel[6], Wolfsburg-Relmhoftz [7], and the NEMO [8] methods. The important point is that any guess which gives a reasonabIe charge distribution is decidedly superior to the use of zero vectors. This will be quantitatively illustrated in the last section after the discussion of the extrapolation method. 169
Yolunie 8,.number‘2 ‘-
;_.
-_
CHEMICALPHYSICSLETTERS
15Januaty1971
Table1
Comparison of the expectation valuei of varlouaopera.tors for the equivalent drbitalandHartree-Fockwavefunctionsof water
-
Expectationvaluea*b) Operator %iF
-%-0 -75.8297'7
LCAOMO -76.00696
cgl
-V/T
2.00202
2.00003
=0 2 *cm
1.4307
1.1744
8,0983
7.2141
5.6817
5.4599
6.8151
6.5122
3Y,2m -em
-1.7750
-1.3132
32zm - +gm
-0.0742
-0.1552
W-0
23.4138
23.4395
l&H
5.5607
s2 cm 2 +rn
-
cc?+1 + z-l
1 cm+1 _Cnt ’ 2 t
(2)
where C?I is taken as zero. The transformation array can be written in the form of an extrapolation table as shown below for five iterations, 4 CY c;
c;
(3)
5.72'70
0.3626
-0.2353
%H/+i
1.8906
1.9576
IH/4
1.4142
1.4595
@q-_r2H)/$
-1.0613
-1.0959
Wj!p-~)/~~
2.1073
2.1'769
9
1056'
2QiSl
W~-r20)/r&
-2.6462
-1.6117
(3+j -Yg,/Yh
-
0.3157
-0.1545
W-H)
0.2140
0.3430
W-0)
292.5101
294.1794
Q90
represent a particular .molecular orbital coefficient as a function of iteration. We then define tile elements of the transfbrniation ‘array as foliows:
a) Only the electroniccontributiona have been given.
b) All resultsare In atomic units.
T&e extrapolation procedure has been dealt with in detail by Shanks [9] and by Wynn [lOJ in connection with divergent and slowly convergent sequences. Petersson and McKay [ll) have also applied.the method to the calculation of molecular integrals. It is‘a non-linear sequence-tosequence transformation denoted as..the eptrans: formation by Shanks. In application to iterative processes, this method is also referred tb as : Aitken’s gencr&ed 6.2process; .‘: .: In order to describe the_ method let the se: ,--_ :. .
The even columns are new sequences of values, each of which converges more rapidly than the previous. Each sequence is transformed according to eq. (2) to produce the following cohnnn. This algorithm for evaluating the transform is due to Wynn [lo]. The element Co is taken as the extrapolated coefficient and save % for use in the next SCF iteration. Because of the oscillatory nature of the molecnlar orbital coefficients as a function of iteration, we have found the B-term transformation to be the most useful in matrix Hartree-Fock calculations. The ek-transformation has been incorporated in our matrix Hartree-Fock routines for a number of years. During this time it has been used in calculations on numerous closed shell molecules ranging in size from first-row Piatpmics to butadiene. In each case the pattern of convergence and the effectiveness of the method were essentially the same. :For this reason we report detailed. information on only one example. UsinglIuzinagals [lZ] .(9s5p)-basis set Ion oxygen and a 3s. basis. scaled to:l.O on ,the hydrogens, a complete.SCF calculation was carri,ed out dn H30. bothwith and_witbo_utktrapolation. : In add$ion, two different.sets’ of starting vectors . \yere.‘y&d f or.,ea;ch. calcGla.i~oii; : Since *e saving8 i- afforded by;_t&eextrapolation depends on the
Volume 8, number 2
The two-electron Iteration 1
2 3 4
5
6 7 8 9 10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25
CBEMICAL PHYSICS LETTERS
15 January1971
Table 2 energy at each iteration for the water ca.IcdatIonsaP)
Case A 69.8998 21.3946 52.9234 26.6997 47.0959 31.6084 42.0668 35.0469 39.5305 36.6700 38.4716 37.3369 38.0496
37.6019 37.8830 37.7065 37.8173 37.7477 37.7914 37.7640 37.7812 37.7704 37.7772 37.7729 37.7756
Case B
37.8441 37.7342 37.8000 37.7505 37.7715 37.7763 37.7735
Case C 37.6138 37.8277 37.7279 37.7994 37.7573 37.7847 37.7679 37.7786 37.7720 37.7762 37.7736 37.7752 37.7742 37.7748 37.7744
Case D
37.7768 37.7735 37.7753 37.7741 37.7748 3717745 37.7746
4
6
s
10
12
14
16
ITERATION
Fig. 1. The total energy of the fcrmaldehyde molecule.
a) Case A: zero vectors and no extrapolations; case B: zero vectors followed by extrapolation of iterations 6-16 and 10-14: case C: EO vectors and no extrapolations; case D: EO vectors followed by extrapolation of iterations 2-6 and 6-10. b) The results are given in atomic units. initial guess, the calculations were started zero vectors as well as the EO’s.
”
with
In order to display the convergence characteristics of the electronic charge distribution, table 2 gives the two-electron energy at each iteration. This quantity is far more sensitive to changes in the wavefunction than is the total energy. The first column lists the two-electron energy for the calculation starting with zero vectors and with no extrapolations. After 25 iterations the total energy has converged to 3 X 10m8 but the two-electron energy has not yet converged past the third decimal place. In the second column the calculation is repeated except that iterations 6-10 and then lo-!4 were transformed according to eq. (2). After the first extrapolation the two-electron, energy is as well converged as that of iteiation 17 in the first column. The two iterations after the second extrapolation are as well converged as iterations 24-25 of the first case; Overall the ko extrapolations .saved about nineriterations..
The third column contains the results for the calculation starting with EO vectors and without extrapolation. After 15 iterations t$z total energy has converged to better than LO . The use of good initial vectors saved more than ten iterations relative to starting with zero vectors. The fourth column gives the results for th? same calculation except that a 5-term transformation was carried out on iterations 2-6 and on iterations 6-10. The first extrapolation saved approximately four iterations and the second extrapolation then converged the two-electron energy to the number of figures quoted. Using one extrapolation and three additional iterations, the energy converged to eight decimal places. This rate of convergence is typical of the calcuiations on other molecules that we have .studied, Le., 8-10 iterations inture convergence of the total energy to about 10’ . In an attempt to apply the method to a divergent case, a calculation on the formaldehyde molecule in a contracted [3s2p/2sl basis was carried out starting with zero vectors:. Such an initial guess appears to be partictiarly inappropriate for formaldehyde. As can be seen in fig. 1, the energy began to diverge, oscillating about a line .with a divergent slope for the first twelve iterations. The vectors from iterations 8-12 were subjected to a 5-term transformation‘and the calculation was conttiued. The energy plotted at iteration 12 was-found using the results of the extrapolation mid represents the antiIimit of the diverging- sequence. .Ttie remaining iterations converge to an energy. of -,144.86495 au. Besides demonstrating a unique feature of the extrapola-
_, ,Xqh+pe 8,‘neber ;. --,_. __-
Z-
._
:, I’
CHErI~A~-P~~IC~-.~E~E~
.-. ._ “:*-tion x&hod, this e&3&e a&o +ws the impor: tancie .of .a ‘good initial guess’ for ,the occupied or.:Tbjtals. . ” : ;.‘.
:
-.-
.~ .._
AC~CNOWLEDGEMEN~
1'
The tiuthors~gratefully aknowledge many. . helpful discuss~qns with fi. George Petersson and Dr. Vintient. McKay. Fitiancial support from department of the California Institute af Technology is &o acknowledged.
_ the -Fhex&stj
REFERENCES.
_
a
[1] ‘X.3. DwuttugJr., J. Chem. Phys. 53 (lS70) 2823.
, .
1
‘_
:-
1
-:
‘15 January 1971
[ii W,R..&&&l, J; Chem. Pi&& 47 (lQ&).32&3. [3] C, C;~;Roothaan and P.S. Bagus, in: Methods in comp&affom&l physfca, Vol. 2, eda. B;Alder, S. Ferabach and M. Rate&erg (Academfa Prass. New : I Yoxk, 1993) p. 47; _[4] yix3ildor ~d_~_~h~Vitt, J* Cbem. Pbys. 44 &66)
j5J J. H:Lktche~ and i. H. DuqningJr., J. C&n. Phye.
, 48 (1968) $538. [6] R.Iiaffman. J. Cbem. Phys. 39 (1964) 1397; 40 (1964) 3247; 2474, 2480. j?] M. WoIfsberg and L.HeImholz, J. Cbem. Pbye. 20 (1952) 837.. [&jM. D. Newt&, F. P. Boer and W. N. Lipscomb, J, Am. Cbem. Sot. 88 (1966).2253, 2361, 2367. [S] D.Sban&t, a.-M&b. Phys. 34 (1955) 1. [lo] P. Wm. Math. Ak?s Computatfon 64 (1956) 91. [ll] G. A. Petersson and V.McKoy, J. Chem. Phys. 46
(1967) 4362.
[li] S.FIuzlna&a,J. Chem. Phys. 42 (l965) 1293,