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An ab initio effective hamiltonian method applied to a relativistic molecule, AgH
Volume 165, number 6
CHEMICAL PHYSICS LETTERS
2 February 1990
AN AB INITIO EFFECTIVE HAMILTONIAN METHOD APPLIED TO A RELATIVISTIC MOLECULE, AgH S. KATSUKI College of General Education, Kyushu University, Fukuoka 810, Japan
Received 2 October 1989; in final form 4 November 1989
An ab initio effective Hamiltonian method for valence electron calculations, where the core-valence interactions and the relativistic effects are expressed in the spectral representation and not in the traditional local potential form is applied to the relativistic molecule AgH. The calculated bond length of 1.702 au, which is 2.5% shorter than the nonrelativistic value, compares well with the Lee-McLean Dirac-Hartree-Fock result of 1.70 au.
1. Introduction Recently an ab initio effective Hamiltonian method has been introduced for valence-electron molecular calculations [ 11. This method is one of the calculation schemes that deal explicitly only with valence electrons, leaving the core electrons in a “frozen” state. Most of the traditional schemes [ 2-51 replace the core-valence interactions by a local and spherically symmetric potential acting on the valence electrons. The potential, which we call the model potential here, is usually fitted to an analytical function prudently so as to reproduce the effects of the core electrons on the valence electrons with high fidelity. This fitting procedure requires a technical skill, which sometimes becomes sublimated to an art, especially when the nonlocal operators, such as an exchange core operator in the Hartree-Fock formalism, are replaced by a local model potential [6]. In the new formalism, on the other hand, the core-valence interactions are expressed in the spectral representation [7-IO] and not in the local potential form. The procedure to get the spectral representation is straightforward and needs no special skill. This is one of the distinguishing traits of the method. Traditional model potential methods assume spherical symmetry of the core-valence interactions and have never considered nonspherical ones. The ab initio effective Hamiltonian method can treat the nonspherically symmetric
core-valence interactions without difficulty. This is another characteristic of the method. In this Letter we report the results of the application of the ab initio effective Hamiltonian method to the relativistic case. The nonspherical treatment of the core-valence interactions will be reported in the near future. In section 2, the ab initio effective Hamiltonian method is reviewed briefly and the formalism of the method as applied to the relativistic case is introduced. The results are described in section 3, with Ag and AgH as examples.
2. An ab initio effective Hamiltonian method The ab initio effective Hamiltonian for n valence electrons of a molecule composed of N atoms is written in atomic units as [l]
H(1,2
,...1
p(i)+ +&MR,J), rb
n)=
isj
(1) with h(i)=
N Z’-Z’
-id,-
1 I
‘Ore+ rli
$ V&,(i) I
+ fj P’(i).
V,J(R,J)_
0009-26 14/90/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland )
(2)
(z’-zcre)w-zore) RIJ
(3)
535
Volume 165, number 6
where Vi,,, represents the effects of the core electrons of the Ith constituent atom working on the valence electrons and is expressed in the present method as Vi,,,(i)
(
=s)’
_
2 February 1990
CHEMICAL PHYSICS LETTERS
zLe
7
+
core 1 [2$(i)-K:(i)] c
I
Q’,
Coulomb and exchange core operators in V’,,, of eq. (4) and the projection operators P’ of eq. (6). Finally, we form the local relativistic potentia1 Vf,, in the spectral representation as follows: V:,,(i) = C @‘( HGv +Hg)Q” /
,
(10)
>
(4) with
(5) where the function set { ]a’>} is a set of primitive Gaussian-type orbitals which constitute the valence orbitals of the Ah constitutent atom. The projection operator P’constructed from the core orbitals @Ewith core energies ei,
where Q” is similar to N of eq. ( 5 ) but the function set { I d’) } in .@’ is a set of primitive Gaussian-type orbitals which constitute the valence orbitals of I symmetry, because the relativistic effects Hhv + Hb are dependent on an orbital index i. We note that no more than one valence orbital for each symmetry I (i=s, p, d, ...) should be included in relativistic molecular calculations. Putting the relativistic operator (eq. (IO) ) into eq. (1 ), we get the ab initio effective Hamiltonian for the relativistic molecular calculations.
core
P’(i)--
c 2tfl@:>(~fl,
(6)
is included in eq. (2) to prevent the valence orbitals from collapsing into the core space. P’could be taken in eq. (4) in the spectral representation form if so desired. The extension of the method to the relativistic case is straightforward if the Cowan-Griffin version [ 111 of the relativistic Hamiltonian, d2 + MI,+l) -du2 I.2
+I’,+!&,+&
G,=t,Gi, (7)
where H,,=-;a’(~,-V,)2, HI,=-~a*[I+~a*(~~-~)]-
is adopted [ 121. First, eq. (7) is solved for all orbitals in the individual atoms. Let {#f} be the core orbitals so obtained for the Ith atom and {$‘u}be the valence orbitals. These orbitals are originally given in numerical form. The next step is to give all these orbitals analytical expression by using appropriate Gaussian-type orbital sets for the sake of analytical molecular calculations. From {#fl we construct the 536
3. Applications
to Ag and AgH
First we estimate the relativistic effect in the Ag atom. After the Cowan-Griffin equation (eq. (7) ) is solved, the relativistic wavefunctions are processed according to the recipe described in section 2. The exponents and coefficients thus obtained are in table 1 in a similar format as that of ref. [ 131. We treat 41~~$zs, $3%, @4%, 42pr @3, and @3d as the core orbitals. The core-valence interactions V,,, calculated with 15 core orbitals and the relativistic effects H,,+ H, obtained from the Cowan-Griffin Hamiltonian are expressed in the spectral representation with use of the primitive Gaussian-type orbitals generated from the ( 16s lop, 7d) set. The relativistic valence orbital energies obtained by the present method are in table 2 together with those obtained by the Cowan-Griffin equation. The orbital energies are slightly dependent on the size of contractions of the primitive Gaussian-type orbitals. The basis set used here is contracted to (10321/721/421) from the (16s lop, 7d) set in table 1. For comparison, table 2 also lists the nonrelativistic orbital energies obtained by the ab initio effective Hamiltonian method of the nonrelativistic version, where J and K in eq. (4) are constructed from the core orbitals of Huzinaga et al. expressed in the (43333/433/43) set [ 13 ] and the basis set for the valence orbitals are
CHEMICAL
Volume 165, number 6
PHYSICS LETTERS
2 February
1990
Table I Exponents and coefficients Orbital symmetry IS
0.98336008 0.05050928 -0.02757692 0.0
Cl
c2 c3 c4 C5
2s
3s
4s
0.36271592 - 1.0476129 -0.01868940 0.00418556
0.15968977 -0.54275519 1.1173844 0.00342522 0.000 16968
0.06682196 -0.23439481 0.61975201 -1.1421464 -0.00377508
s3
s4
0.0
0.0
5s
0.01449668 -0.05726703 0.12414839 -0.39081022
1.0588493
Basis
el e2 e3 e4
Sl
s2
59200.908 6262.0972 1238.3399 322.83881
533.01210 58.168744 24.819341
dl d2 d3 d4
0.018 13260 0.12463336 0.44514169 0.55602871
- 0.09340029 0.62205957 0.4432407
46.363945 7.8107195 3.4563655
S5
6.7140013
0.75563704 0.09899467 0.03566946
1.4372829 0.598285 15
-0.26043897 0.861955822 0.28024552
-0.32004009 0.82703449 0.36625121
-0.07381905 0.59410519 0.48114610
3P
4P
3d
4d
0.45643014 - 1.0708708 - 0.03229862
0.17878750 -0.60625879 1.1339203
P2
P3
I
Orbital symmetry
2P Cl
0.98136012 0.04843457 -0.01656372
c2 c3
PI el e2 e3 e4 dl d2 d3 d4
1741.1513 369.22457 112.17091 39.072669 0.02628723 0.179385 I 1 0.50652847 0.45509124
dl
143.38122 15.802798 5.9238291
2.3011263 0.9346226 0.36046554
-0.02187449 0.50334045 0.57008887
0.47 145302 0.51439522 0.10658127
contracted to (10321/4321/421). Deviations ofthe relativistic orbital energies from the Cowan-Griffin values are l.O%, 1.6Oh and 1.2% for 4p, 4d and 5s orbitals, respectively. The deviations of the orbital energies in the relativistic case are an order of magnitude larger than the nonrelativistic ones. These larger deviations might originate in the errors in the
0.99419968 0.01978730
I
0.31189959 - 1.0422797
d2
158.68116 43.032275 14.608446 5.2261876 0.05155806 0.27847834 0.55251395 0.34693322
3.1986910 0.98841270 0.27260089
0.31850333 0.58486323 0.34329523
fitting process of the wavefunctions and in the evaluation process of the spectral representation of the numerically obtained relativistic potentials. The expectation value of eq. ( 10) taken between the relativistic valence wavefunctions are in table 2. We can thus estimate the physical effects of HMv and HD explicitly owing to our formula (eq. (10)). They are 531
Volume 165, number 6
CHEMICAL PHYSICS LETTERS
2 February 1990
Table 2 Orbital energies and expectation values of relativistic effects for the Ag atom 4P
4d
5s
relativistic this work Cowan-Griffin deviation expect. value
-2.71720 -2.74558 1.0% -0.12660
-0.50312 -0.51150 1.6% -0.01827
-0.23412 - 0.23693 1.2% -0.01436
nonrelativistic this work Huzinaga et al. deviation
-2.65984 -2.65766 0.1%
-0.52099 -0.51935 0.3%
-0.21376 -0.21352 0.1%
Table 3 Molecular constants for AgH
this work Lee-McLean [ 171 McLean [ 18 ] Pyykks[l9] Ziegler et al. [ 201 Martin [21] St011et al. [22] Hay-Martin [ 231 Andzelm et al. [ 241 Ross-Ermler [25] Krauss et al. [ 261 Miyoshi-Sakai [27] expt. [28]
rel.
nonrel.
rel.
nonrel.
rel.
nonrel.
1.702
1.744
1836.4
1712.0
1.46 1.31
1.33
1.733 1.684 1.71 1.763
_
1.70 1.643 1.61 1.694
1.74 1.69 1.61 1.70 1.71 1.74 1.618
1.78 1.65 1.76
usually invisibly tucked into a general potential function [ 14, I5 1. Next we apply the method to the AgH molecule. The relativistic valence basis set is the contracted ( 10321/721/421) set. No polarization function is added. The basis set of Dunning and Hay [ 161 is used for the hydrogen atom. The calculated molecular constants of AgH are presented in table 3 together with nonrelativistic values which are obtained by the ab initio effective Hamiltonian method with use of the basis set of Huzinaga et al. for Ag [ 131 contracted into the same size as those of the relativistic ones. The relativistic and nonrelativistic results compare well with the Dirac-Hartree-Fock result of Lee-Mclean [ 171 and the all-electron result of McLean [ 181. The relativistic effects shorten the 538
0, (eV)
04, (cm-‘)
& (A,
1709 1608 1616 1548 1850 1537 1592 1534 1760
1444 1605 1490 1436 1790 1538
3.07 2.0 1.07 2.0 1.13 2.9 _
1.23 2.69 1.7 0.98 0.98 2.7
0.95 2.39
bond length by 2.5%. This contraction compares with the 4% contraction of the Dirac-Hartree-Fock calculation [ 171. The molecular constants obtained by other studies are also collected in table 3. Table 3 seems to indicate that the present method is useful in relativistic molecular calculations.
Acknowledgement The author would like to express his grateful acknowledgement to Professor Huzinaga of the University of Alberta for his suggestive comments, valuable discussions and encouragement. The numerical calculations were done using the FACOM M780 computing facility of Kyushu University. The pres-
Volume 165, number 6
CHEMICAL PHYSICS LE-lTERS
ent work was supported in part by a Grant-in-Aid for Scientific Research on Priority Area by the Ministry of Education, Science and Culture.
References [ 1 ] S, Katsuki and S. Huzinaga, Chem. Phys. Letters 152 ( 1988) 203. [ 21 R.N. Dixon and 1.L. Robertson, Specialist periodical reports, Vol. 3. Theoretical chemistry (Chem. Sot., London, 1978) p. 100. [ 31 L.R. Kahn, P. Baybutt and D.G. Truhlar, J. Chem. Phys. 65 (1976) 3826. [4] Y. Sakai and S. Huzinaga, J. Chem. Phys. 76 (1982) 2537, 2552. [ 51 S. Katsuki and M. Inokuchi, J. Phys. Sot. Japan 51 ( 1982) 3652. [ 61 Y. Sakai, E. Miyoshi, M. Klobukowski and S. Huzinaga, J. Comput. Chem. 8 ( 1987) 226, 256. [ 71 R.N. Dixon and 1-L.Robertson, Mol. Phys. 37 ( 1979) 1223. [ 8 ] S.C. Leasure, T.P. Martin and G.G. Balint-Kurti, J. Chem. Phys. 80 (1984) 1186. [ 91 J. Andzelm, E. Radzio, 2. Barandiaran and L. Seijo, J. Chem. Phys. 83 (1985) 4565. [ lo] S. Huzinaga, L. Seijo, Z. Barandiaran and M. Klobukowski, J. Chem. Phys. 86 (1987) 2132.
2 February 1990
[ 1 I ] R.D. Cowan and D.C. Griffin, J. Opt. Sot. Am. 66 (1976) 1010. [ 12 ] S. Ratsuki and S. Huzinaga, Chem. Phys. Letters 147 ( I988 ) 597. [ 131 J. Andzelm, M. Klobukowski, E. Radzio-Andzelm, Y. Sakai and H. Tatewaki, in: Gaussian basis sets for molecular calculations, ed. S. Huzinaga (Elsevier, Amsterdam, 1984). [ 141 L.R. Kahn, Intern. J. Quantum Chem. 25 (1984) 149. [ 151M. Klobukowski, J. Comput. Chem. 4 (1983) 350. [ 161T.H. Dunning Jr. and P.J. Hay, in: Modem theoretical chemistry, Vol. 3, ed. H.F. Shaefer III (Plenum Press, New York, 1977) p. 1. [ 171Y.S. Lee and A.D. McLean, J. Chem. Phys. 76 ( 1982) 735. [ 181A.D. McLean, J. Chem. Phys. 79 (1983) 3392. [ 191 P. Pyykkti, J. Chem. Sot. Faraday Trans. 1175 (1979) 1256. [ 201 T. Ziegler, J.G. Snijders and E.J. Baerends, J. Chem. Phys. 74 (1981) 1271. [21]R.L.Martin,J.Phys. Chem. 87 (1983) 750. [ 221 M. Stall, P. Fuentealba, M. Dolg, J. Flad, L. von Szentpaly and H. Preuss, J. Chem. Phys. 79 (1983) 5532. [23] P.J. Hay and R.L. Martin, J. Chem. Phys. 83 (1985) 5174. [24] J. Andzelm, E. Radzio and D.R. Salahub, J. Chem. Phys. 83 (1985) 4573. [25] R.B. Rossand W.C.Ermler, J. Phys. Chem. 89 (1985) 5202. [26] M. Krauss, W.J. Stevens and H. Basch, J. Comput. Chem. 6 (1985) 287. [27] E. Miyoshi and Y. Sakai, J. Comput. Chem. 9 (I 988) 7 19. [28] H. Huber and G. Henberg, Molecular spectra and molecular structure. Vol. 4. Constants of diatomic molecules (Van Nostrand Reinhold, New York, 1979).
539
CHEMICAL PHYSICS LETTERS
2 February 1990
AN AB INITIO EFFECTIVE HAMILTONIAN METHOD APPLIED TO A RELATIVISTIC MOLECULE, AgH S. KATSUKI College of General Education, Kyushu University, Fukuoka 810, Japan
Received 2 October 1989; in final form 4 November 1989
An ab initio effective Hamiltonian method for valence electron calculations, where the core-valence interactions and the relativistic effects are expressed in the spectral representation and not in the traditional local potential form is applied to the relativistic molecule AgH. The calculated bond length of 1.702 au, which is 2.5% shorter than the nonrelativistic value, compares well with the Lee-McLean Dirac-Hartree-Fock result of 1.70 au.
1. Introduction Recently an ab initio effective Hamiltonian method has been introduced for valence-electron molecular calculations [ 11. This method is one of the calculation schemes that deal explicitly only with valence electrons, leaving the core electrons in a “frozen” state. Most of the traditional schemes [ 2-51 replace the core-valence interactions by a local and spherically symmetric potential acting on the valence electrons. The potential, which we call the model potential here, is usually fitted to an analytical function prudently so as to reproduce the effects of the core electrons on the valence electrons with high fidelity. This fitting procedure requires a technical skill, which sometimes becomes sublimated to an art, especially when the nonlocal operators, such as an exchange core operator in the Hartree-Fock formalism, are replaced by a local model potential [6]. In the new formalism, on the other hand, the core-valence interactions are expressed in the spectral representation [7-IO] and not in the local potential form. The procedure to get the spectral representation is straightforward and needs no special skill. This is one of the distinguishing traits of the method. Traditional model potential methods assume spherical symmetry of the core-valence interactions and have never considered nonspherical ones. The ab initio effective Hamiltonian method can treat the nonspherically symmetric
core-valence interactions without difficulty. This is another characteristic of the method. In this Letter we report the results of the application of the ab initio effective Hamiltonian method to the relativistic case. The nonspherical treatment of the core-valence interactions will be reported in the near future. In section 2, the ab initio effective Hamiltonian method is reviewed briefly and the formalism of the method as applied to the relativistic case is introduced. The results are described in section 3, with Ag and AgH as examples.
2. An ab initio effective Hamiltonian method The ab initio effective Hamiltonian for n valence electrons of a molecule composed of N atoms is written in atomic units as [l]
H(1,2
,...1
p(i)+ +&MR,J), rb
n)=
isj
(1) with h(i)=
N Z’-Z’
-id,-
1 I
‘Ore+ rli
$ V&,(i) I
+ fj P’(i).
V,J(R,J)_
0009-26 14/90/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland )
(2)
(z’-zcre)w-zore) RIJ
(3)
535
Volume 165, number 6
where Vi,,, represents the effects of the core electrons of the Ith constituent atom working on the valence electrons and is expressed in the present method as Vi,,,(i)
(
=s)’
_
2 February 1990
CHEMICAL PHYSICS LETTERS
zLe
7
+
core 1 [2$(i)-K:(i)] c
I
Q’,
Coulomb and exchange core operators in V’,,, of eq. (4) and the projection operators P’ of eq. (6). Finally, we form the local relativistic potentia1 Vf,, in the spectral representation as follows: V:,,(i) = C @‘( HGv +Hg)Q” /
,
(10)
>
(4) with
(5) where the function set { ]a’>} is a set of primitive Gaussian-type orbitals which constitute the valence orbitals of the Ah constitutent atom. The projection operator P’constructed from the core orbitals @Ewith core energies ei,
where Q” is similar to N of eq. ( 5 ) but the function set { I d’) } in .@’ is a set of primitive Gaussian-type orbitals which constitute the valence orbitals of I symmetry, because the relativistic effects Hhv + Hb are dependent on an orbital index i. We note that no more than one valence orbital for each symmetry I (i=s, p, d, ...) should be included in relativistic molecular calculations. Putting the relativistic operator (eq. (IO) ) into eq. (1 ), we get the ab initio effective Hamiltonian for the relativistic molecular calculations.
core
P’(i)--
c 2tfl@:>(~fl,
(6)
is included in eq. (2) to prevent the valence orbitals from collapsing into the core space. P’could be taken in eq. (4) in the spectral representation form if so desired. The extension of the method to the relativistic case is straightforward if the Cowan-Griffin version [ 111 of the relativistic Hamiltonian, d2 + MI,+l) -du2 I.2
+I’,+!&,+&
G,=t,Gi, (7)
where H,,=-;a’(~,-V,)2, HI,=-~a*[I+~a*(~~-~)]-
is adopted [ 121. First, eq. (7) is solved for all orbitals in the individual atoms. Let {#f} be the core orbitals so obtained for the Ith atom and {$‘u}be the valence orbitals. These orbitals are originally given in numerical form. The next step is to give all these orbitals analytical expression by using appropriate Gaussian-type orbital sets for the sake of analytical molecular calculations. From {#fl we construct the 536
3. Applications
to Ag and AgH
First we estimate the relativistic effect in the Ag atom. After the Cowan-Griffin equation (eq. (7) ) is solved, the relativistic wavefunctions are processed according to the recipe described in section 2. The exponents and coefficients thus obtained are in table 1 in a similar format as that of ref. [ 131. We treat 41~~$zs, $3%, @4%, 42pr @3, and @3d as the core orbitals. The core-valence interactions V,,, calculated with 15 core orbitals and the relativistic effects H,,+ H, obtained from the Cowan-Griffin Hamiltonian are expressed in the spectral representation with use of the primitive Gaussian-type orbitals generated from the ( 16s lop, 7d) set. The relativistic valence orbital energies obtained by the present method are in table 2 together with those obtained by the Cowan-Griffin equation. The orbital energies are slightly dependent on the size of contractions of the primitive Gaussian-type orbitals. The basis set used here is contracted to (10321/721/421) from the (16s lop, 7d) set in table 1. For comparison, table 2 also lists the nonrelativistic orbital energies obtained by the ab initio effective Hamiltonian method of the nonrelativistic version, where J and K in eq. (4) are constructed from the core orbitals of Huzinaga et al. expressed in the (43333/433/43) set [ 13 ] and the basis set for the valence orbitals are
CHEMICAL
Volume 165, number 6
PHYSICS LETTERS
2 February
1990
Table I Exponents and coefficients Orbital symmetry IS
0.98336008 0.05050928 -0.02757692 0.0
Cl
c2 c3 c4 C5
2s
3s
4s
0.36271592 - 1.0476129 -0.01868940 0.00418556
0.15968977 -0.54275519 1.1173844 0.00342522 0.000 16968
0.06682196 -0.23439481 0.61975201 -1.1421464 -0.00377508
s3
s4
0.0
0.0
5s
0.01449668 -0.05726703 0.12414839 -0.39081022
1.0588493
Basis
el e2 e3 e4
Sl
s2
59200.908 6262.0972 1238.3399 322.83881
533.01210 58.168744 24.819341
dl d2 d3 d4
0.018 13260 0.12463336 0.44514169 0.55602871
- 0.09340029 0.62205957 0.4432407
46.363945 7.8107195 3.4563655
S5
6.7140013
0.75563704 0.09899467 0.03566946
1.4372829 0.598285 15
-0.26043897 0.861955822 0.28024552
-0.32004009 0.82703449 0.36625121
-0.07381905 0.59410519 0.48114610
3P
4P
3d
4d
0.45643014 - 1.0708708 - 0.03229862
0.17878750 -0.60625879 1.1339203
P2
P3
I
Orbital symmetry
2P Cl
0.98136012 0.04843457 -0.01656372
c2 c3
PI el e2 e3 e4 dl d2 d3 d4
1741.1513 369.22457 112.17091 39.072669 0.02628723 0.179385 I 1 0.50652847 0.45509124
dl
143.38122 15.802798 5.9238291
2.3011263 0.9346226 0.36046554
-0.02187449 0.50334045 0.57008887
0.47 145302 0.51439522 0.10658127
contracted to (10321/4321/421). Deviations ofthe relativistic orbital energies from the Cowan-Griffin values are l.O%, 1.6Oh and 1.2% for 4p, 4d and 5s orbitals, respectively. The deviations of the orbital energies in the relativistic case are an order of magnitude larger than the nonrelativistic ones. These larger deviations might originate in the errors in the
0.99419968 0.01978730
I
0.31189959 - 1.0422797
d2
158.68116 43.032275 14.608446 5.2261876 0.05155806 0.27847834 0.55251395 0.34693322
3.1986910 0.98841270 0.27260089
0.31850333 0.58486323 0.34329523
fitting process of the wavefunctions and in the evaluation process of the spectral representation of the numerically obtained relativistic potentials. The expectation value of eq. ( 10) taken between the relativistic valence wavefunctions are in table 2. We can thus estimate the physical effects of HMv and HD explicitly owing to our formula (eq. (10)). They are 531
Volume 165, number 6
CHEMICAL PHYSICS LETTERS
2 February 1990
Table 2 Orbital energies and expectation values of relativistic effects for the Ag atom 4P
4d
5s
relativistic this work Cowan-Griffin deviation expect. value
-2.71720 -2.74558 1.0% -0.12660
-0.50312 -0.51150 1.6% -0.01827
-0.23412 - 0.23693 1.2% -0.01436
nonrelativistic this work Huzinaga et al. deviation
-2.65984 -2.65766 0.1%
-0.52099 -0.51935 0.3%
-0.21376 -0.21352 0.1%
Table 3 Molecular constants for AgH
this work Lee-McLean [ 171 McLean [ 18 ] Pyykks[l9] Ziegler et al. [ 201 Martin [21] St011et al. [22] Hay-Martin [ 231 Andzelm et al. [ 241 Ross-Ermler [25] Krauss et al. [ 261 Miyoshi-Sakai [27] expt. [28]
rel.
nonrel.
rel.
nonrel.
rel.
nonrel.
1.702
1.744
1836.4
1712.0
1.46 1.31
1.33
1.733 1.684 1.71 1.763
_
1.70 1.643 1.61 1.694
1.74 1.69 1.61 1.70 1.71 1.74 1.618
1.78 1.65 1.76
usually invisibly tucked into a general potential function [ 14, I5 1. Next we apply the method to the AgH molecule. The relativistic valence basis set is the contracted ( 10321/721/421) set. No polarization function is added. The basis set of Dunning and Hay [ 161 is used for the hydrogen atom. The calculated molecular constants of AgH are presented in table 3 together with nonrelativistic values which are obtained by the ab initio effective Hamiltonian method with use of the basis set of Huzinaga et al. for Ag [ 131 contracted into the same size as those of the relativistic ones. The relativistic and nonrelativistic results compare well with the Dirac-Hartree-Fock result of Lee-Mclean [ 171 and the all-electron result of McLean [ 181. The relativistic effects shorten the 538
0, (eV)
04, (cm-‘)
& (A,
1709 1608 1616 1548 1850 1537 1592 1534 1760
1444 1605 1490 1436 1790 1538
3.07 2.0 1.07 2.0 1.13 2.9 _
1.23 2.69 1.7 0.98 0.98 2.7
0.95 2.39
bond length by 2.5%. This contraction compares with the 4% contraction of the Dirac-Hartree-Fock calculation [ 171. The molecular constants obtained by other studies are also collected in table 3. Table 3 seems to indicate that the present method is useful in relativistic molecular calculations.
Acknowledgement The author would like to express his grateful acknowledgement to Professor Huzinaga of the University of Alberta for his suggestive comments, valuable discussions and encouragement. The numerical calculations were done using the FACOM M780 computing facility of Kyushu University. The pres-
Volume 165, number 6
CHEMICAL PHYSICS LE-lTERS
ent work was supported in part by a Grant-in-Aid for Scientific Research on Priority Area by the Ministry of Education, Science and Culture.
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