- Email: [email protected]
More on the open-shell problem
Volume
70, number
CHEMKXL
2
PHYSICS
LETTERS
1 MarcIs E980
MORE ON THE OPEN-SHELL PROBLEM Josip HENDEKOVJC, “‘RudJer BoSkovri” Recetved
3 December
Davor KIRIN
and Mihca PAVLOVLC
Instmtute, 41001 Zagreb, 0oatra,
Yugoslavia
1979
The Iterative algorithm Forsolvmgtheopenshell SCF secular equattons proposed earher, is tested numerkaBY ou several states of ozone A strong dependence of the convergence rate on the values of the Free parameters $rll is revealed. The a&onthm is Found to be compatible with Davidson’s chorce of these parameters. It is shown also how this algorithm I'IUY be USed
when the cilculatron rs performed III a nonilrthogonal
basrs set.
1. Introduction
2. Analysis of the procedure
One of the present authors (JH) had recently proposed an iterative algorrthm [l] for solvmg the set of coupled open-shell SCF equations [Z] , after they are formally decoupled by the method of Huzinaga [3] _ The algorithm is constructed on the assumption that the foliowmg relation is fulfU.led
Several open-shell states of ozone were analyzed numerically. The open-chain conformation with the internuclear angle B = 1 16.S” and the O-O bond length R = 2.4126 au was chosen. The cakulations were performed wrth a double-zeta gaussian lobe basis set as m ref. [4]. The ground-state energy of -224.2356 hartree was obtained for the configuratiora (. 6af ia$4b$ lb:)_ Several open-shell states, Iisted in table 1, were considered. The main aim of the numerical analysis was to show how the convergence rate depends on the value of the parameter hPq, starting with the closed-shell SCF solutions as initial vectors. The iterative algorithm
co where Q@) stands for decoupled SCF matrices transformed mto the basrs set consisting of the SCF solutions, eq. (10) of ref. [l]. Kp stands for the index set specifyrng orbrtals in shell “p”. From the following identity, eq. (15) of ref. [ 11,
Table 1 it IS clear that condition (I) is fuhiied provided the undetermmed parameters hp4, which appear in the Huzinaga procedure, are chosen to be symmetnc, hp4 = X@. In thrs letter numerical evidence is given about the convergence rate of the proposed Iterative rdgorithm. It is ako shown that the requirement for the parameters Xp4 to be symmetric is not essential for the vahdrty of this algorithm. Finally we explam how the algonthm is apphed when the calculation is performed in the non-orthogonal basrs set.
Number
=
of iterations required
to reach the convergenQ
(Ae
10s6 hartree) for several states of ozone, and for three values
oFh=‘hpq State
Energy &artree)
Number of iterations A = 0.5
n=i*o
h=L5
3At(6at
--* 7ar)
-223.8761
34
26
33
3A2(4bl
-c 2bt)
-224.2084
73
41
29
3Br (6ar + 2br )
-224.2108
64
37
27
‘Aa(4b2
--c 2bl)
-224.2889
57
30
23
3Ba(iaa
+ 2br)
-224.3419
43
33
‘Br(6ar
-r 2bt)
-224.1878
73
41
29 >50
379
VoIume 70, number 2
I March 1980
CHEMICAL PHYSICS LETTERS
of ref. [l] was apphed without any averaging or other convergence accelerating procedure. The calculations were ended when the energy difference between two successrve steps was less than low6 hartree. From table I it IS seen that Xw = I .O is a relatively good choxce for ail states considered, although it reduces the coupling operator Xqp in eq. (6) of ref. [I] to a single term XqJJ = F(q). For the state lA2 we studied the convergence more carefully by choosing several values for Xl, = h21. The obtained results are presented in fig, I. To our surprise we found exactly the same energy and the same rate of convergence with Davidson’s nonsymmetric choice of parameters j\pq: h21 = 1.0, Al, = 0.0, ref. [Sl , as with Xl2 = X21 = 1.0, although we beheved that our rteratrve procedure was not suitable for such a chotce. Closer mspection reveals, however, that with the non-symmetric XPs, condrtton (1) is still satrsfied due to an inherent feature of the Iterative algorithm itself. Namely, from fig 1b of ref. [l] It follows that Q$$ = 0 d 0~E KP, fl E Kq and p
h,,~cppl(F(P) - Fql~J
= 0,
Thus, rf hqP f 0 for p < 4, our iterative algorithm guarantees [due to the rdentrty (2)] that condition (1) is satisfied. This justifies the use of the proposed iterative procedure also for non-symmetric XP4, provided Xqp#Owhenp
As shown in fig. 1 the convergence rate may depend very much on the choice of parameters XPc _We certa.mIy cannot give the best universal prescrlption for those free parameters. The symmetnc choice hPq = 1 .O-1.5, or the non-symmetric prescription of Davrdson seem to be a reasonable first try.
3. Non-orthogonal
basis set
If the basis set is non-orthogonal one starts with the followmg set of equations, written in components, mstead of eq. (1) of ref. [l] :
-0.175
Frg. 1 Dependenceof tbe ’ Aa energy on the numberof iterationsN, for severalvaluesof the parameter h (II = k&. The reIative where E = (-224 0 + A) hartree.
energies A are plotted,
380
Vohut~e 70. number 2
CHEMICAL PHYSICS LETTERS
index 6 goes over all non-empty orbitals. The transformation matrix u, which is not orthogona! now, and the overlap matrix S are defined by
Here Ix,) are non-orthogonal basis functions. The probIem of solving the open-shell SCF secular equatrons (4) may be reduced to the same problem in the orthogonal basis set by the following transformation of eq. (4)_ 7EfCp, where Fjjp) = (S-1/2 Ffp)!S-‘~2)r,,
1 March 1980
4. ConcIusions We have demonstrated above that the convergence of the ite,ative procedure for solving the open-shell SCF secular equations, proposed in ref. [I] ; depends very much on the choice of parameters hp4_ From several numeric examples considered here, we anticiQ to lie between 2.0 and pate the optimal values of 1.5. Contrary to the on’ A* assumption in ref- CIj we found that-h,, need not be symmetric. Additional convergence accelerating techniques can be implemented to the present iterative algorithm in the usual way. When working wrth a non~~ho~on~ basis set this algorithm may be used without any modification provided the original SCF equations are transformed as in eq. (6).
m
U& = c (s1/2),1 ulk* (7) I=1
Eq. (6) has exactly the same form as eq. (1) of ref. [I] when written in components, and the matrix u KI eq. (7) IS orthogonal, so that the iterative algorithm constructed in ref. [I] applies also to eq. (6), derived above.
References [ 11 J. Hendekovic, Theoret. Cbim. Acta 42 (1976) 193.
[2] C.C.J. Roothsan. Rev. Mod. Phys 32 (1960) 179.
[3] S. Huzinaga, J. Chem. Phys. 51 (1969) 3971. (41 S. Shih, RJ. Buenker and SD_ Peyerimhoff. Chem. Phys.
Letters 28 f&974) 463. [5 J E-R. DaMson, Chem. Phys. Letters 21 (1973) 56%
38k
70, number
CHEMKXL
2
PHYSICS
LETTERS
1 MarcIs E980
MORE ON THE OPEN-SHELL PROBLEM Josip HENDEKOVJC, “‘RudJer BoSkovri” Recetved
3 December
Davor KIRIN
and Mihca PAVLOVLC
Instmtute, 41001 Zagreb, 0oatra,
Yugoslavia
1979
The Iterative algorithm Forsolvmgtheopenshell SCF secular equattons proposed earher, is tested numerkaBY ou several states of ozone A strong dependence of the convergence rate on the values of the Free parameters $rll is revealed. The a&onthm is Found to be compatible with Davidson’s chorce of these parameters. It is shown also how this algorithm I'IUY be USed
when the cilculatron rs performed III a nonilrthogonal
basrs set.
1. Introduction
2. Analysis of the procedure
One of the present authors (JH) had recently proposed an iterative algorrthm [l] for solvmg the set of coupled open-shell SCF equations [Z] , after they are formally decoupled by the method of Huzinaga [3] _ The algorithm is constructed on the assumption that the foliowmg relation is fulfU.led
Several open-shell states of ozone were analyzed numerically. The open-chain conformation with the internuclear angle B = 1 16.S” and the O-O bond length R = 2.4126 au was chosen. The cakulations were performed wrth a double-zeta gaussian lobe basis set as m ref. [4]. The ground-state energy of -224.2356 hartree was obtained for the configuratiora (. 6af ia$4b$ lb:)_ Several open-shell states, Iisted in table 1, were considered. The main aim of the numerical analysis was to show how the convergence rate depends on the value of the parameter hPq, starting with the closed-shell SCF solutions as initial vectors. The iterative algorithm
co where Q@) stands for decoupled SCF matrices transformed mto the basrs set consisting of the SCF solutions, eq. (10) of ref. [l]. Kp stands for the index set specifyrng orbrtals in shell “p”. From the following identity, eq. (15) of ref. [ 11,
Table 1 it IS clear that condition (I) is fuhiied provided the undetermmed parameters hp4, which appear in the Huzinaga procedure, are chosen to be symmetnc, hp4 = X@. In thrs letter numerical evidence is given about the convergence rate of the proposed Iterative rdgorithm. It is ako shown that the requirement for the parameters Xp4 to be symmetric is not essential for the vahdrty of this algorithm. Finally we explam how the algonthm is apphed when the calculation is performed in the non-orthogonal basrs set.
Number
=
of iterations required
to reach the convergenQ
(Ae
10s6 hartree) for several states of ozone, and for three values
oFh=‘hpq State
Energy &artree)
Number of iterations A = 0.5
n=i*o
h=L5
3At(6at
--* 7ar)
-223.8761
34
26
33
3A2(4bl
-c 2bt)
-224.2084
73
41
29
3Br (6ar + 2br )
-224.2108
64
37
27
‘Aa(4b2
--c 2bl)
-224.2889
57
30
23
3Ba(iaa
+ 2br)
-224.3419
43
33
‘Br(6ar
-r 2bt)
-224.1878
73
41
29 >50
379
VoIume 70, number 2
I March 1980
CHEMICAL PHYSICS LETTERS
of ref. [l] was apphed without any averaging or other convergence accelerating procedure. The calculations were ended when the energy difference between two successrve steps was less than low6 hartree. From table I it IS seen that Xw = I .O is a relatively good choxce for ail states considered, although it reduces the coupling operator Xqp in eq. (6) of ref. [I] to a single term XqJJ = F(q). For the state lA2 we studied the convergence more carefully by choosing several values for Xl, = h21. The obtained results are presented in fig, I. To our surprise we found exactly the same energy and the same rate of convergence with Davidson’s nonsymmetric choice of parameters j\pq: h21 = 1.0, Al, = 0.0, ref. [Sl , as with Xl2 = X21 = 1.0, although we beheved that our rteratrve procedure was not suitable for such a chotce. Closer mspection reveals, however, that with the non-symmetric XPs, condrtton (1) is still satrsfied due to an inherent feature of the Iterative algorithm itself. Namely, from fig 1b of ref. [l] It follows that Q$$ = 0 d 0~E KP, fl E Kq and p
h,,~cppl(F(P) - Fql~J
= 0,
Thus, rf hqP f 0 for p < 4, our iterative algorithm guarantees [due to the rdentrty (2)] that condition (1) is satisfied. This justifies the use of the proposed iterative procedure also for non-symmetric XP4, provided Xqp#Owhenp
As shown in fig. 1 the convergence rate may depend very much on the choice of parameters XPc _We certa.mIy cannot give the best universal prescrlption for those free parameters. The symmetnc choice hPq = 1 .O-1.5, or the non-symmetric prescription of Davrdson seem to be a reasonable first try.
3. Non-orthogonal
basis set
If the basis set is non-orthogonal one starts with the followmg set of equations, written in components, mstead of eq. (1) of ref. [l] :
-0.175
Frg. 1 Dependenceof tbe ’ Aa energy on the numberof iterationsN, for severalvaluesof the parameter h (II = k&. The reIative where E = (-224 0 + A) hartree.
energies A are plotted,
380
Vohut~e 70. number 2
CHEMICAL PHYSICS LETTERS
index 6 goes over all non-empty orbitals. The transformation matrix u, which is not orthogona! now, and the overlap matrix S are defined by
Here Ix,) are non-orthogonal basis functions. The probIem of solving the open-shell SCF secular equatrons (4) may be reduced to the same problem in the orthogonal basis set by the following transformation of eq. (4)_ 7EfCp, where Fjjp) = (S-1/2 Ffp)!S-‘~2)r,,
1 March 1980
4. ConcIusions We have demonstrated above that the convergence of the ite,ative procedure for solving the open-shell SCF secular equations, proposed in ref. [I] ; depends very much on the choice of parameters hp4_ From several numeric examples considered here, we anticiQ to lie between 2.0 and pate the optimal values of 1.5. Contrary to the on’ A* assumption in ref- CIj we found that-h,, need not be symmetric. Additional convergence accelerating techniques can be implemented to the present iterative algorithm in the usual way. When working wrth a non~~ho~on~ basis set this algorithm may be used without any modification provided the original SCF equations are transformed as in eq. (6).
m
U& = c (s1/2),1 ulk* (7) I=1
Eq. (6) has exactly the same form as eq. (1) of ref. [I] when written in components, and the matrix u KI eq. (7) IS orthogonal, so that the iterative algorithm constructed in ref. [I] applies also to eq. (6), derived above.
References [ 11 J. Hendekovic, Theoret. Cbim. Acta 42 (1976) 193.
[2] C.C.J. Roothsan. Rev. Mod. Phys 32 (1960) 179.
[3] S. Huzinaga, J. Chem. Phys. 51 (1969) 3971. (41 S. Shih, RJ. Buenker and SD_ Peyerimhoff. Chem. Phys.
Letters 28 f&974) 463. [5 J E-R. DaMson, Chem. Phys. Letters 21 (1973) 56%
38k