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The split hamiltonian matrix method and the interaction between two hydrogen atoms
Volume 9. number 4
THE AND
15 Mny 1971
CHEMICAL PHYSICS LETTERS
THE
SPLIT
HAMILTONIAN
INTERACTION
MATRIX
BETWEEN
TWO
METHOD
HYDROGEN
ATOMS
K. PECUL Quantum Cltcrnisty Gvoup, Institute ct~~b’usic Problems of Citemisty. Uniuersity of Warsaw, Pash?n 1, Warsaw 22, Polaand
Received 15 March 1971
The hnmiltonian matrix in the Ritz variational equation js split into the 9mpcrturbedtt and “perturbed” parts. and now mstsix equations are derived. The Pauli principle and symmetry are properly taken into :tccount, The cnkulstcd energies of the intern&ion between the two hydrogen atoms are in good ngreement with the varintional results.
Let us consider the calculation of the total approximate e?~rgy, E, by using the Ritz variational method with Iinear parameters Cj in the finite n-dimensional basis. The column vector, C, composed of these Cj, is the solutton of the equation (w-ES)C
=o ,
0)
H and S are the hamiltonian and the Gram (of overlap) matriceS, respectively. Let us assume that
where
H=U
+P
,
C=C%A,,
E=E%W e iz,
where P, A 3 fz ;;r$ $,
W, are small in comparison
with
~~~e,Eti~l~~~~~~~s~~~e~d
to be real and symmetkic matrices. Since in our method the hamiltonian matrix (and not the hamiltonian operator) is spiit, the name proposed for this method is “the split h~iltoni~ matrix method”. The U, CO and ~0 fulfil (U-EOS)GO=
D,
(3)
where D is the remainder vector, small in comSuch a vector is used by parison with A, Roothaan and Bagus [l] in their variaticnal mothod. If cqs. (2) and (3) are combined with eq. (l), the result is: (U-~S}Ae=-(P-W~S)G”-D
- (P-WeS)Ae* (41
The second-order term ( P- WeS)Ae dropped, eq. (4) is reduced to
316
being
(U - EOS)A = -(P-WS)CO
- D.
(5)
Since eq. (5), contrary to eq. (4), is not equivalent to eq. (2), the symbols A, W, instead of Ae, We, respectively, are used in eq. (5). By multipl ing eq. (5), on the left, by the row vector GoJ and neglecting the term LIT-A we get w=cOTPCO’
because C0T.D = 0. If D is not very small, we can obtain A directly from eqs. (5) and (6). However, if D is very small (or zero-vector), the matrix ( U - BOS)‘is practically (or exactly) singular. In such a case we can solve eq. (5) by using the method described in ref. [l] i.e. assuming at the first stage the component Al = 0, calculating other components from eq. (5), with the first equation being removed and the restriction that Al = 0, and finally calculating A1 from the condition AT-CO = 0. The vector A is an exact solution of eq. (5) when D = 0. Whenever D + 0, the vector A is only an approximate nolution; in the case of a very firnaIl D, however, it is numeri‘tally exact. The mean value of the hamiltonian operator in the finite basis can be expressed, after some manipulations, by E = ~(2) t (L+A%A)-1
(-ATPGO(A%A)
+ AT.(zD-WA)+ATPA),
where E(2)=_&‘+W+ATPC6. When Ujm = 0 and Sjm = 0, given 1 ,C j f k, and k C m ,C ?a, then eq. (3) splits out into two
(6)
Volume 9. number 1
CHEMICAL
block equations. Let a few lowest eigenvalues So of eq. (3) belong to the “upper” block, i.e. the
The symbol R denotes the internucleu distance. The obtained values of E(2) and E, given in 10-a au, are shown in tabIe 1. These values are shifted by 1 au, i,e. by the energy of the two scparated hydrogen atoms. Therefore, the numbers given in table 1 are simply the interaction energies. The basis used in the calculations is suc-
components Cy of their eeenvectors vanish for k K j G R. If we calculate W, A and B corrc-
spending to such an eigenvalue Eo, the Pjln for k < (j, m) C n ‘are present only in tine term ATPA . Therefore, if we determine E(2) only, we do not need to calculate those Pjnl. Our method can also be appiied to the excited states when the e’ envectors belonging to the lower eigenvaluesE % , of a given symmetry, tie eliminated in a way similar to the Sinanoglu procedure [2]. The numerical calculations were carried out ior the two interacting hydrogen atoms in X 1 $ and b3Ci states at R = 8, 10 and 12 au. The antisymmetrized products of the Slater-type orbitals with common 1s hydrogen atom orbital exponent were used in the n-dimensional basis. The total wavefunction is of the correct symmetry with respect to the exchange of electrons and protons. The EO and CO represent the v‘ariational solution of es.(l) in a k-dimensional basis, k cn. They are calculated by the variational Ostrowski method modified by the present author in such a way that the resulting interaction energy is calculated directly, and not as the difference of the much larger total energies (see ref.[3] where the details are provided). The equations il.,,- -0 and Sj,,- -0 for 1 < j c k and k < 1~ < H are fu JO lllcd since the respective one-center parts of Hjp)z are zero, and by the Schmidt orthonormalization procedure, respectively. The elements of the matrix
cessively built up with the folIowing, EIeitIerLondon typo, 12 configurations: Lsls, 2~02~0, 2~12~1. 2~03~0. 2~13~1, &oZdo, QxW, Wo4f0, 2p14fl, 3d03d0, 3d13dl, 3d23d2. In order to ob-
tain a C state each configuration with the positive “magnetic” quantum number is combined together with that of the corresponding negative nunlber e.g. the 2~131~1configuration with the 2p_13p_1 configuration. Therefore, if e.g. the numbers IZ,~ in the first column of table 1 are equal to 10.5, it means that the IO-dimensional basis is used, the last configuration included being 3dO3d0, and that the “unperturbed* function contains 5 terms, i.e. all the lsfs, 2p2p and 2p3p configurations, the last configuration included being 2~13~1. The values of i? are given in table 1 only at R = 8 au, because at R = 10 au ‘andR 12 au, and for any k, they are equa1 to the variational (k = n) interaction energies in all the digits listed. The results show a good agreement of perturbational and variational interaction cnergies, especially at R = I2 au. It seems that at larger R the agreement should be better. The interaction energies calculated with the D vector given by eq. (3), and those obtained under the assumption that D = 0 are equal in all the digits listed. Since even simple calculations as carried out here iead TV reliable results, one may expect that at least for some interaction problems our method will prove more useful than other pertur-
u are Ujm,
=
forl~(j,m)~k;
IIjm
for 1 c j G k ,
=o = $_m,
Ifj,l
for&<(j,m)G
16 Mny 1971
PHYSICS LETTERS
k c m c ,I ;
n.
Table 1 Internctionencrgios in 10-8 au. R in au
n, k
X1X+ 6
b
R=12
It=10
R=8
E (2)
Em
E (2)
-~___._______-___._-_ 10.10 -249.1
-839.5
__- _..._ -.-5033
R=12
R-8 i3
E(2)
_ _.__- -.---5033 -246.3
3x; R=6
R=lO
EW E(2) ..._ _._.-__-__-----.-.---
R=s
B
-739.2
-17S2
-1782
10, 7
-249.1
-839.5
-5033
-5033
-246.3
-739.3
-1782
-1782
10, 5
-249.1
-839.0
-5030
-5032
-246.3
-739.8
-1789
-1780
10. 1
-248.8
-837.1
-5020
-5032
-2.16.2
-741.9
-1850
-1778
12.12
-844.3
-5075
-5075
-743.7
-1812
-1812
12. 7
-844.3
-5076
-5075
-743.8
-L81.&
-1812
12. 5
-843.8
-5072
-5075
-74% .3
-L82P
-1811
12, 1
-842.4
-5063
-5074
-746.8
-1887
-1808 _-A-
.---.--
--
_-_
3’7
bational approaches.
Further details pertaining
to the method and its possible
improvements
will be presented eIsewhere. The author feels fessor Wtidzimierz ing of the manuscript.
318
15 May 1971
CHE&ffWL PHYSICS LETTERS
Volume 9, number 4
very much indebted to ProKolos for his critical read-
REFERENCES [Z] C. C.J. ~oo~~~n nnd P. S. Bagus, Methods in com-
putational physics, Vol. 2 {Academic Press, New York, 1963)p. 47. fS] 0. Sinano~lu. Phys.Rev. 122 (1961)491. [3] K. Pecul, Acta Phys. Polon. A39 {1971) 9.
7 . d
THE AND
15 Mny 1971
CHEMICAL PHYSICS LETTERS
THE
SPLIT
HAMILTONIAN
INTERACTION
MATRIX
BETWEEN
TWO
METHOD
HYDROGEN
ATOMS
K. PECUL Quantum Cltcrnisty Gvoup, Institute ct~~b’usic Problems of Citemisty. Uniuersity of Warsaw, Pash?n 1, Warsaw 22, Polaand
Received 15 March 1971
The hnmiltonian matrix in the Ritz variational equation js split into the 9mpcrturbedtt and “perturbed” parts. and now mstsix equations are derived. The Pauli principle and symmetry are properly taken into :tccount, The cnkulstcd energies of the intern&ion between the two hydrogen atoms are in good ngreement with the varintional results.
Let us consider the calculation of the total approximate e?~rgy, E, by using the Ritz variational method with Iinear parameters Cj in the finite n-dimensional basis. The column vector, C, composed of these Cj, is the solutton of the equation (w-ES)C
=o ,
0)
H and S are the hamiltonian and the Gram (of overlap) matriceS, respectively. Let us assume that
where
H=U
+P
,
C=C%A,,
E=E%W e iz,
where P, A 3 fz ;;r$ $,
W, are small in comparison
with
~~~e,Eti~l~~~~~~~s~~~e~d
to be real and symmetkic matrices. Since in our method the hamiltonian matrix (and not the hamiltonian operator) is spiit, the name proposed for this method is “the split h~iltoni~ matrix method”. The U, CO and ~0 fulfil (U-EOS)GO=
D,
(3)
where D is the remainder vector, small in comSuch a vector is used by parison with A, Roothaan and Bagus [l] in their variaticnal mothod. If cqs. (2) and (3) are combined with eq. (l), the result is: (U-~S}Ae=-(P-W~S)G”-D
- (P-WeS)Ae* (41
The second-order term ( P- WeS)Ae dropped, eq. (4) is reduced to
316
being
(U - EOS)A = -(P-WS)CO
- D.
(5)
Since eq. (5), contrary to eq. (4), is not equivalent to eq. (2), the symbols A, W, instead of Ae, We, respectively, are used in eq. (5). By multipl ing eq. (5), on the left, by the row vector GoJ and neglecting the term LIT-A we get w=cOTPCO’
because C0T.D = 0. If D is not very small, we can obtain A directly from eqs. (5) and (6). However, if D is very small (or zero-vector), the matrix ( U - BOS)‘is practically (or exactly) singular. In such a case we can solve eq. (5) by using the method described in ref. [l] i.e. assuming at the first stage the component Al = 0, calculating other components from eq. (5), with the first equation being removed and the restriction that Al = 0, and finally calculating A1 from the condition AT-CO = 0. The vector A is an exact solution of eq. (5) when D = 0. Whenever D + 0, the vector A is only an approximate nolution; in the case of a very firnaIl D, however, it is numeri‘tally exact. The mean value of the hamiltonian operator in the finite basis can be expressed, after some manipulations, by E = ~(2) t (L+A%A)-1
(-ATPGO(A%A)
+ AT.(zD-WA)+ATPA),
where E(2)=_&‘+W+ATPC6. When Ujm = 0 and Sjm = 0, given 1 ,C j f k, and k C m ,C ?a, then eq. (3) splits out into two
(6)
Volume 9. number 1
CHEMICAL
block equations. Let a few lowest eigenvalues So of eq. (3) belong to the “upper” block, i.e. the
The symbol R denotes the internucleu distance. The obtained values of E(2) and E, given in 10-a au, are shown in tabIe 1. These values are shifted by 1 au, i,e. by the energy of the two scparated hydrogen atoms. Therefore, the numbers given in table 1 are simply the interaction energies. The basis used in the calculations is suc-
components Cy of their eeenvectors vanish for k K j G R. If we calculate W, A and B corrc-
spending to such an eigenvalue Eo, the Pjln for k < (j, m) C n ‘are present only in tine term ATPA . Therefore, if we determine E(2) only, we do not need to calculate those Pjnl. Our method can also be appiied to the excited states when the e’ envectors belonging to the lower eigenvaluesE % , of a given symmetry, tie eliminated in a way similar to the Sinanoglu procedure [2]. The numerical calculations were carried out ior the two interacting hydrogen atoms in X 1 $ and b3Ci states at R = 8, 10 and 12 au. The antisymmetrized products of the Slater-type orbitals with common 1s hydrogen atom orbital exponent were used in the n-dimensional basis. The total wavefunction is of the correct symmetry with respect to the exchange of electrons and protons. The EO and CO represent the v‘ariational solution of es.(l) in a k-dimensional basis, k cn. They are calculated by the variational Ostrowski method modified by the present author in such a way that the resulting interaction energy is calculated directly, and not as the difference of the much larger total energies (see ref.[3] where the details are provided). The equations il.,,- -0 and Sj,,- -0 for 1 < j c k and k < 1~ < H are fu JO lllcd since the respective one-center parts of Hjp)z are zero, and by the Schmidt orthonormalization procedure, respectively. The elements of the matrix
cessively built up with the folIowing, EIeitIerLondon typo, 12 configurations: Lsls, 2~02~0, 2~12~1. 2~03~0. 2~13~1, &oZdo, QxW, Wo4f0, 2p14fl, 3d03d0, 3d13dl, 3d23d2. In order to ob-
tain a C state each configuration with the positive “magnetic” quantum number is combined together with that of the corresponding negative nunlber e.g. the 2~131~1configuration with the 2p_13p_1 configuration. Therefore, if e.g. the numbers IZ,~ in the first column of table 1 are equal to 10.5, it means that the IO-dimensional basis is used, the last configuration included being 3dO3d0, and that the “unperturbed* function contains 5 terms, i.e. all the lsfs, 2p2p and 2p3p configurations, the last configuration included being 2~13~1. The values of i? are given in table 1 only at R = 8 au, because at R = 10 au ‘andR 12 au, and for any k, they are equa1 to the variational (k = n) interaction energies in all the digits listed. The results show a good agreement of perturbational and variational interaction cnergies, especially at R = I2 au. It seems that at larger R the agreement should be better. The interaction energies calculated with the D vector given by eq. (3), and those obtained under the assumption that D = 0 are equal in all the digits listed. Since even simple calculations as carried out here iead TV reliable results, one may expect that at least for some interaction problems our method will prove more useful than other pertur-
u are Ujm,
=
forl~(j,m)~k;
IIjm
for 1 c j G k ,
=o = $_m,
Ifj,l
for&<(j,m)G
16 Mny 1971
PHYSICS LETTERS
k c m c ,I ;
n.
Table 1 Internctionencrgios in 10-8 au. R in au
n, k
X1X+ 6
b
R=12
It=10
R=8
E (2)
Em
E (2)
-~___._______-___._-_ 10.10 -249.1
-839.5
__- _..._ -.-5033
R=12
R-8 i3
E(2)
_ _.__- -.---5033 -246.3
3x; R=6
R=lO
EW E(2) ..._ _._.-__-__-----.-.---
R=s
B
-739.2
-17S2
-1782
10, 7
-249.1
-839.5
-5033
-5033
-246.3
-739.3
-1782
-1782
10, 5
-249.1
-839.0
-5030
-5032
-246.3
-739.8
-1789
-1780
10. 1
-248.8
-837.1
-5020
-5032
-2.16.2
-741.9
-1850
-1778
12.12
-844.3
-5075
-5075
-743.7
-1812
-1812
12. 7
-844.3
-5076
-5075
-743.8
-L81.&
-1812
12. 5
-843.8
-5072
-5075
-74% .3
-L82P
-1811
12, 1
-842.4
-5063
-5074
-746.8
-1887
-1808 _-A-
.---.--
--
_-_
3’7
bational approaches.
Further details pertaining
to the method and its possible
improvements
will be presented eIsewhere. The author feels fessor Wtidzimierz ing of the manuscript.
318
15 May 1971
CHE&ffWL PHYSICS LETTERS
Volume 9, number 4
very much indebted to ProKolos for his critical read-
REFERENCES [Z] C. C.J. ~oo~~~n nnd P. S. Bagus, Methods in com-
putational physics, Vol. 2 {Academic Press, New York, 1963)p. 47. fS] 0. Sinano~lu. Phys.Rev. 122 (1961)491. [3] K. Pecul, Acta Phys. Polon. A39 {1971) 9.
7 . d