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Accelerating the convergence of the quadratically convergent SCF method
Votume 36, number 5
CHEMCAL
ACCELEEZATMG THE CONVERGENCE
PHYSICS LETTERS
I5 December
OF THE QUADRATICALLY
1975
CONVERGENT SCF METHOD
LC. CHANG *
Received
17 June 1975
An algorithm
to the calculation original method.
to solve
the divergence problem of the quadratically convergent SCF method is suggested nnd is applied of the excited states of a number of small molecu!es. The resuits show definite imp:ovement over tic
In the past decade, the quadratically convergent SCF method was freydently used to so!ve the correct SCF equations of open shell systems [l-S]. The method is generally considered to be more numericalLy stable than the open shell SCF theory formulated by Roothaan [6] and extended to more general forms by Huzinaga [7] ; however, it also displays divergence in many cases. The precise conditions under which the method would diverge are unknown, hut the divergence usually OCCUL when the initial orbit& are inadequate to give a reasonable charge distribution. The usual procedure of solving this problem . is to terminate the iteration and start with a new set of trial orbitals. The procedure may be repeated many times and there is no guarantee that such a set of orbit& which leads to convergence can be obtained by trial and error. Recently, we have successfully applied an algorithm to overcome such a difficulty, and the enere always converges after a minimal number of iterations. The idea of the aIgoritIun to be presented owes its origmity to the work of Fletcher and Powell 181, and Bruner and Moemer [9]. To afford the stability and to accelerate the convergence, several changes ir the generation of a new set of trial orbitals have been made. Instead of using the equation sugested by Goddard and co-workers [I], the orbitals are generated by * Present address: Department of Chemistry, University, Tempe, Arizo~ 85281. USA.
Arizona
State
(1) The sign and the step size h of the correction coefticients, Caij), are chosen at the values at thigh the function I?@) E@) =E(#
f- ha&
0)
is a minimum, so that the overall minimization of energy is’cnrried out by a sequence of one-dimensional searches. The correction coefficients, A+ are evaluated by the equations as sumsted by Goddard and coworkers [ 11,
(3) where B$ = 2 [(_4ik-A,k-AgfAj,)
{li’l kl)
+(b,k-b,,-Cli,+&j~)(ikljl)]
tS~~(il~-~II)
-6i/(kl~-i$lj)
+6ik
Qii = M&qjl assuming that all the orbitals are real. At the beginning of each iteration, the step k is set to be ur-&-j (X = l), and the sign is chosen such that the first derivative of..&h), E’(Q
(4) 3 (5)
: Volume 36, number
5
CHEMICAL PHYSICS LETTERS
Table 1
Table
hiolecular gometriesa)
15 December 1975
2
used in the LxlcuIations
Results of small molecuIesa) Molecule
Tiolecule
Synimetiy
Bond len$hs
WI2
Czh
Rcc
HCN
c,
C2H4-
Dzd
CHzNz
czv
OF2
Czv
WHO
Cs
(A)
Bond nngks (deg)
State
Nuniber of iterations I
= 1.3;36
RCH RC.J &-H RCC RCH RC.J RNN
= 1.1086 = 1.31113 = 1.1!57 = 1.3761 = 1.1:!24 = 1.3541 = 1.2001 = 1.1067 QH ROF = 1.5233 = 1.2163 Rco RCHr = 1.1331 @I2 = 1.2520 -
LHCC
= 135.3
LHCFI
= 132.5
LHCH
= 106.2
cz I12 HCN CzH4 CH2N2
LHCH
= 12i.5
HCHO
LFOF = 78 LOCH1 = 14.3 LOCH:! = 104
a) ‘The geometries of HCN, C*K& and C2& are INDO predicted geometries, whereas those of CH2N2, OF2 and HCHO
were set to the v;llues at wl~icl~ divergence
was
found.
is less than zero. IfE(I) E(O), E@) is expanded as 2 fifth degree of polynomial in X,
and a value of X, h’, between 0 and 1 which corresto a minimum of the pol-+nomial can be found. The values of the C,r’s can be easily o5iained from the equations of&X), E’(J), E”@) at X = 0 and 1. If E(l) is still larger than E(O), the interval is halved and the search for an optimal value of X continues. The iteration is then reReated until the final convergence is reached. To test its usefulness, we applied the present ~algorithrr to the excited states of a number of small ponds
molecules, HCN, C,H,, C,b, Cb_N,, OF2 and ECHO, within the framework o,f the IhA approximation. For comparison, iye also carried out calcula-
tions using the original version of the quadratically convergent &S
SCF method
in which the step size was
to unity.-The getimetries
of those molccuies are ‘listed in- table 1. The initial orbit* were obtained ir’iom the simple Hiickel calcula,lion. Suc_h an initial gue&qpears td be rather inappropriate for CI-12Na,
OF2
‘AU(~-n*) ‘A” rBs(n-rr*) ‘Al ‘BP 3A’(n- cr*)
Converged enegy (au)
II
5 5 6 no convergence no convergence no convergence
-26.666431 -29.310496 -35.408362 -61.310496 -112.692412 -42.719313
8 9 12
a) I, the original algorithm (A = 1); II, this algoriihm (A is varied when the iteration diverges).
OF, and HCHO. As referred to table 2, the ori@al algorithm converges rapidly only for the first three cases (HCN, CzH2, C2H4) but diverges for CH2N2, GF2
and HCHO.
[lo,1
I]
Several
extrapolation
techniques
were employed ii1order to overcome the
divergence problem, bat none was successful. However, if the energy was expanded as a polynomial of X and was mi_nimized with respect to X after the second iteration, an optimal value of the step X could be &t&-red. The set of improved orbitals generated by such an optimal value of X(x = 0.458 for CH,N, and 0.477 for OF,) did lead the iteration to convergence. Even in the worst condition, such as the case of the 3A’ state (n-o*) of HCHO, two more additional interpolations of X are required before the fmd convergence is reached. The results presented
iteration by iteration are listed in table 3. From the above samples, it can be easily noted that the algorithm suggested here is exactly the same as that suggested in ref. [I] as long as the iteration converges. If the energy value begins to increase, several steps to prevent the divergence are made. The decrease of the energy is thus guaranteed at each iteration and an optimal set of orbitals should be obtained eventwlly. The algorithrr not only retains the quadratic convergence property of the original method but also affords its stability. The algorithm should be useful
for stiidies
of the potential
surfaces
r?f mol-
ecules since the usual SCF methods do not converge properly if the geometry is far from the equilibrium gecmetry. Although the sample calculation is carried
Volume 36, number 5 Table 3 Results of Cl32 N2,OF2,
15 December 1975
CHEMICAL PHYSICS LETTERS
and IiCHCI
OF2
CH&
HCHO
Iteration I
II
I
II
I
I1
2
-61.40591 -61.39873
-112.62522 - 11 X62439
-61.33760 -61.28654 -61.20154
-112.61656 -112.47890 -112.17623
-112.62522 -112.62439 (A = 9.477) -112.65156 -112.67741 -112.68718
-42.67094 -42.59929
3 4 5
-61.40591 -61.39873 (h = 0.458) -61.41345 --61.42434 -61.43012
-4Z.G~~i.z -42.55929 (h=O.125) -42.69650 -42.71 S83
6
-61.11352 -61.00431 no convergence
-6lS3143 -61.43145 -6:.43145
-112.04717 -112.00534 no con-
1
7 8 9 10
-112.69139
-112.69239 -112.69241
-42.58762 -42.47655 -42.35341
-a2.71291 (h = 0.089)
-42.28543 -42.16286 no con-
-42.‘; 189.5 -42.71928 -42.71407 (ix= O.M3f -42.71330
vergence
vergence
-112.69241
conversd
-42
converged
7193’
-42:7193; converged
11
12 -out within
the INDO approximation,
the algorithm
should also be useful for other quantum
mechanical
approximations. The author
wishes to express his gratitude to Dr.
B-L. Bruner for sugyxting the problem.
References [ 11 1V.J. iknt,
W.A. Goddard_111 2nd T.H. S)unnin~~Chem. Phys. Letters 6 (1970) 147.
Suppl.Progr.Theoret. Phrs. 40 (1967) 37. J. Masse, Cahiers Phys. 1.5 (1961) 453. W.R. Wcssel, J. Chen Phys. 47 (1967) 3253. S. Huzinag, Rrogr. Theoret. Phys. 41 (1969) 507. C.C.J. Roothnan, Rev. Mod. Phys. 37, (1960) 179. S. tiuzinaga, Phys. Rcu. I20 f196i3) 866. R Fletcher and 1M.J.D. Powell, Computek J. 6 (1963) 163. WA, Bioemer and B.L. Bruner, J. Chem. Phys. 58 (1973) 3735. C.CJ. Ro~thaan and P.S. Bagus, Methods Comp. Phys. 2 (1963) 47. N.W. Winter and T.H. Dunning, Chem. Fhys. Letters 8 (1971) 169.
13 J. Hinze and C.C,J. Roothaan,
CHEMCAL
ACCELEEZATMG THE CONVERGENCE
PHYSICS LETTERS
I5 December
OF THE QUADRATICALLY
1975
CONVERGENT SCF METHOD
LC. CHANG *
Received
17 June 1975
An algorithm
to the calculation original method.
to solve
the divergence problem of the quadratically convergent SCF method is suggested nnd is applied of the excited states of a number of small molecu!es. The resuits show definite imp:ovement over tic
In the past decade, the quadratically convergent SCF method was freydently used to so!ve the correct SCF equations of open shell systems [l-S]. The method is generally considered to be more numericalLy stable than the open shell SCF theory formulated by Roothaan [6] and extended to more general forms by Huzinaga [7] ; however, it also displays divergence in many cases. The precise conditions under which the method would diverge are unknown, hut the divergence usually OCCUL when the initial orbit& are inadequate to give a reasonable charge distribution. The usual procedure of solving this problem . is to terminate the iteration and start with a new set of trial orbitals. The procedure may be repeated many times and there is no guarantee that such a set of orbit& which leads to convergence can be obtained by trial and error. Recently, we have successfully applied an algorithm to overcome such a difficulty, and the enere always converges after a minimal number of iterations. The idea of the aIgoritIun to be presented owes its origmity to the work of Fletcher and Powell 181, and Bruner and Moemer [9]. To afford the stability and to accelerate the convergence, several changes ir the generation of a new set of trial orbitals have been made. Instead of using the equation sugested by Goddard and co-workers [I], the orbitals are generated by * Present address: Department of Chemistry, University, Tempe, Arizo~ 85281. USA.
Arizona
State
(1) The sign and the step size h of the correction coefticients, Caij), are chosen at the values at thigh the function I?@) E@) =E(#
f- ha&
0)
is a minimum, so that the overall minimization of energy is’cnrried out by a sequence of one-dimensional searches. The correction coefficients, A+ are evaluated by the equations as sumsted by Goddard and coworkers [ 11,
(3) where B$ = 2 [(_4ik-A,k-AgfAj,)
{li’l kl)
+(b,k-b,,-Cli,+&j~)(ikljl)]
tS~~(il~-~II)
-6i/(kl~-i$lj)
+6ik
Qii = M&qjl assuming that all the orbitals are real. At the beginning of each iteration, the step k is set to be ur-&-j (X = l), and the sign is chosen such that the first derivative of..&h), E’(Q
(4) 3 (5)
: Volume 36, number
5
CHEMICAL PHYSICS LETTERS
Table 1
Table
hiolecular gometriesa)
15 December 1975
2
used in the LxlcuIations
Results of small molecuIesa) Molecule
Tiolecule
Synimetiy
Bond len$hs
WI2
Czh
Rcc
HCN
c,
C2H4-
Dzd
CHzNz
czv
OF2
Czv
WHO
Cs
(A)
Bond nngks (deg)
State
Nuniber of iterations I
= 1.3;36
RCH RC.J &-H RCC RCH RC.J RNN
= 1.1086 = 1.31113 = 1.1!57 = 1.3761 = 1.1:!24 = 1.3541 = 1.2001 = 1.1067 QH ROF = 1.5233 = 1.2163 Rco RCHr = 1.1331 @I2 = 1.2520 -
LHCC
= 135.3
LHCFI
= 132.5
LHCH
= 106.2
cz I12 HCN CzH4 CH2N2
LHCH
= 12i.5
HCHO
LFOF = 78 LOCH1 = 14.3 LOCH:! = 104
a) ‘The geometries of HCN, C*K& and C2& are INDO predicted geometries, whereas those of CH2N2, OF2 and HCHO
were set to the v;llues at wl~icl~ divergence
was
found.
is less than zero. IfE(I)
and a value of X, h’, between 0 and 1 which corresto a minimum of the pol-+nomial can be found. The values of the C,r’s can be easily o5iained from the equations of&X), E’(J), E”@) at X = 0 and 1. If E(l) is still larger than E(O), the interval is halved and the search for an optimal value of X continues. The iteration is then reReated until the final convergence is reached. To test its usefulness, we applied the present ~algorithrr to the excited states of a number of small ponds
molecules, HCN, C,H,, C,b, Cb_N,, OF2 and ECHO, within the framework o,f the IhA approximation. For comparison, iye also carried out calcula-
tions using the original version of the quadratically convergent &S
SCF method
in which the step size was
to unity.-The getimetries
of those molccuies are ‘listed in- table 1. The initial orbit* were obtained ir’iom the simple Hiickel calcula,lion. Suc_h an initial gue&qpears td be rather inappropriate for CI-12Na,
OF2
‘AU(~-n*) ‘A” rBs(n-rr*) ‘Al ‘BP 3A’(n- cr*)
Converged enegy (au)
II
5 5 6 no convergence no convergence no convergence
-26.666431 -29.310496 -35.408362 -61.310496 -112.692412 -42.719313
8 9 12
a) I, the original algorithm (A = 1); II, this algoriihm (A is varied when the iteration diverges).
OF, and HCHO. As referred to table 2, the ori@al algorithm converges rapidly only for the first three cases (HCN, CzH2, C2H4) but diverges for CH2N2, GF2
and HCHO.
[lo,1
I]
Several
extrapolation
techniques
were employed ii1order to overcome the
divergence problem, bat none was successful. However, if the energy was expanded as a polynomial of X and was mi_nimized with respect to X after the second iteration, an optimal value of the step X could be &t&-red. The set of improved orbitals generated by such an optimal value of X(x = 0.458 for CH,N, and 0.477 for OF,) did lead the iteration to convergence. Even in the worst condition, such as the case of the 3A’ state (n-o*) of HCHO, two more additional interpolations of X are required before the fmd convergence is reached. The results presented
iteration by iteration are listed in table 3. From the above samples, it can be easily noted that the algorithm suggested here is exactly the same as that suggested in ref. [I] as long as the iteration converges. If the energy value begins to increase, several steps to prevent the divergence are made. The decrease of the energy is thus guaranteed at each iteration and an optimal set of orbitals should be obtained eventwlly. The algorithrr not only retains the quadratic convergence property of the original method but also affords its stability. The algorithm should be useful
for stiidies
of the potential
surfaces
r?f mol-
ecules since the usual SCF methods do not converge properly if the geometry is far from the equilibrium gecmetry. Although the sample calculation is carried
Volume 36, number 5 Table 3 Results of Cl32 N2,OF2,
15 December 1975
CHEMICAL PHYSICS LETTERS
and IiCHCI
OF2
CH&
HCHO
Iteration I
II
I
II
I
I1
2
-61.40591 -61.39873
-112.62522 - 11 X62439
-61.33760 -61.28654 -61.20154
-112.61656 -112.47890 -112.17623
-112.62522 -112.62439 (A = 9.477) -112.65156 -112.67741 -112.68718
-42.67094 -42.59929
3 4 5
-61.40591 -61.39873 (h = 0.458) -61.41345 --61.42434 -61.43012
-4Z.G~~i.z -42.55929 (h=O.125) -42.69650 -42.71 S83
6
-61.11352 -61.00431 no convergence
-6lS3143 -61.43145 -6:.43145
-112.04717 -112.00534 no con-
1
7 8 9 10
-112.69139
-112.69239 -112.69241
-42.58762 -42.47655 -42.35341
-a2.71291 (h = 0.089)
-42.28543 -42.16286 no con-
-42.‘; 189.5 -42.71928 -42.71407 (ix= O.M3f -42.71330
vergence
vergence
-112.69241
conversd
-42
converged
7193’
-42:7193; converged
11
12 -out within
the INDO approximation,
the algorithm
should also be useful for other quantum
mechanical
approximations. The author
wishes to express his gratitude to Dr.
B-L. Bruner for sugyxting the problem.
References [ 11 1V.J. iknt,
W.A. Goddard_111 2nd T.H. S)unnin~~Chem. Phys. Letters 6 (1970) 147.
Suppl.Progr.Theoret. Phrs. 40 (1967) 37. J. Masse, Cahiers Phys. 1.5 (1961) 453. W.R. Wcssel, J. Chen Phys. 47 (1967) 3253. S. Huzinag, Rrogr. Theoret. Phys. 41 (1969) 507. C.C.J. Roothnan, Rev. Mod. Phys. 37, (1960) 179. S. tiuzinaga, Phys. Rcu. I20 f196i3) 866. R Fletcher and 1M.J.D. Powell, Computek J. 6 (1963) 163. WA, Bioemer and B.L. Bruner, J. Chem. Phys. 58 (1973) 3735. C.CJ. Ro~thaan and P.S. Bagus, Methods Comp. Phys. 2 (1963) 47. N.W. Winter and T.H. Dunning, Chem. Fhys. Letters 8 (1971) 169.
13 J. Hinze and C.C,J. Roothaan,