CI calculation on the Rydberg spectrum of H3

Volume 168, number 1

CHEMICAL PHYSICS LETTERS

20 April 1990

CI CALCULATION ON THE RYDBERG SPECTRUM OF Ha G.H.F. DIERCKSEN, W. DUCH * and J. KARWOWSIU ’ Max-Planck-Institutjiir PhysikundAstrophysik,Institutjiir Astrophysik, Karl-Schwarzschiid-Strasse 1,8046 GarchingnearMunich, FederalRepublicof Germany Received 16 January 1990

Energies for the ground state and for eight excited states of the triatomic hydrogen molecule in its equilateral geometry with a bond length of 1.633 bobr have been obtained by large-scale multi-reference configuration-interaction (CI) calculations using basis sets of contracted Gaussian functions. The effects on the resulting spectrum due to the change of the basis set and the manyparticle wavefunction have been studied. In particular it has been shown that the sequence of energy levels depends on the size of the basis set. The calculated excitation energies are in good agreement with experimenttal data.

1. Introduction The triatomic hydrogen molecule is the prototype system of the class of polyatomic Rydberg molecules [ 11. It meets all requirements defining these interesting and difficult to study systems: the ground state of the neutral molecule is repulsive, there is a large energy gap between the ground and the first excited state [I] and the corresponding positive ion is stable. The first excited state of HS already has Rydberg character, i.e. it may be described with relatively good accuracy as the state of one electron associated with a diffuse orbital moving in a core consisting of two electrons and three protons. The spectrum of this molecule has been studied both experimentally [ 25] and theoretically [ 6-101. The theoretical studies on the H3 spectrum have been based on either Hartree-Fock calculations [7-91 or have used rather limited configuration-interaction expansions [ 1Cl1, in contrast to highly advanced calculations on the ground states of both the H3 molecule [ 1 l-151 and its positive ion [ 16-201. This study has two aims: ( 1) to check how sensitive are the results of CI molecular Rydberg-state calculations to the choice of the ’ Permanent address: Zakfad Informatyki Stosowanej, Uniwersytet Mikolaja Kopemika, Grudzi&ka 5, Ton& Poland. 2 Permanent address: Instytut Fizyki, Uniwersytet Mikotaja Kopemika, Grudzhdzka 5, Ton& Poland.

orbital basis set and (2) to check how important are the electron-correlation effects in predicting the locations of the molecular Rydberg states. We have performed large scale CI calculations, using several sets of one-electron basis functions, including rather extended basis sets.

2. Details of the calculations 2.1. Gaussianbasissets Five basis sets of Gaussian orbitals have been used in this study: ( 1) The basis number 4 of King and Morokuma [ 61. This basis set consists of 98 orbitals: 4 contracted s- and 2 p-type orbitals at each of the protons and 2 s-, 8 p- and 7 d-type orbitals at the center of the molecule. The orbitals at the molecular center describe the Rydberg states and were very carefully selected for an accurate description of the 15 lowest Rydberg states of the H3 molecule at the one-electron level [ 61. (2) The basis of Dykstra et al. [ 161. This 63-orbital basis consists of 6 contracted s-type, 3 p-type and 1 d-type orbit& centered at each of the protons. It was constructed for determining a very accurate ground-state energy of the H3+ ion. ( 3 ) The basis of Dykstra ( 2 ) augmented by 23 or-

0009.2614/90/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland)

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(2 s-, 3 p- and 2 d-type) at the center of the molecule. This augmented Dykstra basis consists of 86 orbitals. The exponents have been partly optimized to give the lowest values for the first Rydberg states of HS. The exponents are 0.118 and 0.003 for thes-,0.128,0.064andO.O32forthep-,2.9and0.01 for the d-type functions. (4) The final basis of Siegbahn and Liu [ 111. The basis consists of 57 contracted orbitals: 4 s-, 3 p-, and 1 d-type at each proton. It was designed to obtain very accurate ground-state energies of the H, molecule. ( 5 ) The basis of Siegbahn and Liu (4) augmented by 38 orbitals (2 s-, 6 p- and 3 d-type) at the center of the molecule. These 38 orbitals have been obtained through a reduction of the set of Rydberg orbitals of Ring and Morokuma [6] followed by partial optimization of the exponents. The final values of the exponents are: 0.118 and 0.03 1 for the s-, 0.160, 0.080, 0.040, 0.020, 0.010 and 0.005 for the p-,0.100,0.033 and 0.010 for the d-type orbitals. The resulting basis consists of 95 orbitals. bitals

2.2. Many-particle wavefunctions

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for the ground state of the H3+ ion in its equilateral geometry with the bond length equal to 1.633 bohr, close to its equilibrium bond length [ 1,6,21]. The configuration-interaction calculations have been performed by taking either all SCF orbitals or only that part of them, corresponding to the lowest orbital energies. In the case of the g&function basis ( 1) the number of orbit& used in the CI calculation has been limited to 5 1. This kind of SCF basis-set reduction has been designated as 98 J 51. In the cases of the 86-, 57- and 95-function basis sets (3 ), (4) and (5), respectively, the complete and several reduced basis sets have been used. The ground-state energies of the H: ion have been obtained by full CI calculations within both the complete and the reduced basis sets. In the case of the HS molecule, the ground and excited-state energies have been obtained by either full CI calculations or by calculations taking all singly and doubly excited states from a multi-reference space consisting of an internal space containing the most important configurations, in general the complete active space of 7-9 orbitals. The error relative to the full CI results caused by this restriction of the manyparticle wavefunction is expected to be smaller than the error due to truncating the SCF orbital basis set.

The Hartree-Fock SCF equations have been solved

Table 1 Ground-state energies of Hf obtained using the one-determinant approximation (&) mation @et). All energies arc in millihartrees Gaussian basis set designation

number of orbitals

(1)

98151 63 86 57 95118 95129 95144 95156 p95168 95184 95

(2) (3) (4) (5)

HF limit [ 6 ] correlated limit (Hylleraas CI) [ 201

70

Number of CI basis functions

&F

447 682 1246 573 66 157 348 547 762 1102 1455

-

and the full configuration-interaction approxi-

E MIT

&I

(=&IF-&I)

1299.91 1300.28 1300.29 1300.28 f 300.29

-

1310.45 1342.72 1342.90 1342.48 1307.06 1310.62 1319.70 1330.83 1338.54 1341.65 1342.51

10.54 42.44 42.61 42.20 6.77 10.33 19.41 30.54 38.25 41.36 42.22

- 1343.50

43.30

- 1300.36

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CHEMICAL PHYSICS LEITERS

Volume 168, number 1

3. Results 3.1. Ground-stateenergies of H$ and H3 The ground-state energies of the H: ion obtained for different basis sets in the one-determinant and in the full CI approximations, are collected in table 1. As can be seen in all cases the SCF energies are close to the Hartree-Fock limit, estimated to be - 1300.36 mhartree [6]. The full CI energies vary between - I3 10.5 mhartree in the case of basis set ( 1) and - 1342.5 to - 1342.9 mhartree in the cases of the complete basis sets (2) to (5), compared to the estimated value of the exact energy of 1343.5 mhartree [20]. It is interesting to note that the lowest 51 orbitals of basis set ( 1) create a CI space which accounts only for about 25% of the exact correlation energy estimated to be 43.30 mhartree [ 201. On the other hand all Gaussian basis sets designed to describe accurately the ground-state energy (basis sets (2)- (5) ) account for 97.5%-98.4% of the exact correlation energy. The orbitals describing the Rydberg

spectrum have very small (if any) influence on the calculated correlation energy. Augmenting basis set (2) by adding 23 orbitals and basis set (4) by adding 38 orbitals improves the calculated correlation energy only by 0.02 mhartree, i.e. by 0.05Oh. The results for the ground state of the Hj molecule are displayed in table 2. Again a very strong dependence of the correlation energy on the one-electron basis set rather than on the length of the CI expansion is clearly observed. The relation between the correlation energy obtained forH3+ and H3 and the basis sets used is shown in fig. 1. The behaviour of the correlation energy versus the number of basis functions for both systems is very regular and similar for all bases considered. Though the curves in fig. 1 show saturation of the basis set, the limit value of the correlation energy of H3 determined in this study is estimated to be x 5 mhartree smaller than the exact value.

Table 2 Ground-state energies of Hs obtained using the one-determinant approximation (&r ) and the configuration-interaction method (&,). All eneraies are in millihartrees CI basis

Gaussian basis set designation

number of orbitals

number of internal orbit&

number of CI *) basis functions

(1)

98151 63 86126 86135 86141 86148 86163 86171 57&22 57130 57150 57 95118 95129 95144 95~56 95168 95

51 8 26 35 41 48 8 8 9 4 4 4 10 10 10 10 10 10

11022 9681 1693 4252 6494 11704 9357 11940 1131 800 2117 2843 970 3280 8810 15290 22570 47060

(2) (3)

(4)

(5)

- 1496.96 - 1496.96 - 1497.38

-1496.16

- 1497.33

‘) Number of A, functions of the Czv subgroup of Ds symmetry in case of basis sets (l)-(4) subgroup of Ds symmetry in case of basis set ( 5) .

-1511.52 - 1559.95 - 1522.31 - 1537.58 - 1547.39 - 1555.60 - 1557.72 - 1558.38 - 1547.23 - 1551.88 - 1556.95 - 1557.59 - 1505.80 -1511.86 - 1527.19 - 1543.75 - 1554.30 - 1558.97

14.56 61.99 24.93 40.21 50.01 58.27 60.34 61.00 51.07 55.72 60.79 61.43 8.47 14.53 29.86 46.42 56.97 61.64

and number of A, functions of the Cz

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K Fig. 1.Ground-state correlation energy of Hs and H: calculated using CI expansions based on the orbital spaces(4) 57 J (57-K) (triangles) and (5 ) 95 1 ( 95 -K) (dots) versus K.

20 April 1990

3d-type density distributions of the Rydberg electron. In case of the one-electron basis set ( 1) consisting of 5 1 orbitals the CzVsubgroup of D3,,has been used to factorize the Hamiltonian matrix. The CI basis contains all configurations (full CI). In case of basis set (5 ) the subgroup C2 of DBhhas been used instead. This means, that A, and Bz states of C2,,correspond to A states of Cz and similarly At and B, states of CzVcorrespond to B of C2. The CI basis for each of the two symmetry species of C2 consists of the full internal space spanned by 10 orbitals and of all configurations which are singly or doubly excited relative to this space. The set of internal orbitals is composed of the lowest a{ orbital (doubly occupied by the core electrons in the one-determinant approximation) and the lowest 9 orbitals of the studied symmetry. The corresponding CI basis consists of 47060 spin eigenstates in the case of A symmetry and of 48660 in the case of B symmetry. An inspection of table 3 shows that the sequence of states in the calculated spectrum depends strongly on the electron correlation. In particular, the Hartree-Fock sequence

3.2. Excited states

E(3s)cE(3p,)cE(3d,)
To discuss the basis set dependence of the HJ spectrum the differences between the ground-state energy of Hz and the energies of the lowest 9 levels of H3 calculated in basis ( 1) and in different subsets of basis (5) are displayed in table 3. The states considered are corresponding to the 2s-, 2p-, 3s-, 3p- and

is changed to

provided the CI space is sufficiently large. The correlation-energy contributions defined as

Table 3 Differences between the Cl energies of the 9 lowest states of Hr and the full Cl ground-state energy of Hz in different orbital basis sets. For the specification of the CI bases, see text. All energies are in millihartrees Orbital basis sets D3,

1 2E’ 1 *A’I

1 *A; 2 2E’ 2 2A; 2 ‘A; 3 *E’ 1 2E” 3 *A;

72

orbital type 2Px, 2P, 2s 2Pz 3Px, 3P, 3s 3Pz 3d, 3d,z_,s 3dxn 3d,z 3d,z

C2” AI,

B2

A, BI A,,

B2

Al B, 4,

B2

B,,

A2

A*

(I)98151

(5)95129

(5)95144

(5)95156

(5195168

(5195

201.07 138.08 129.81 71.69 58.66 56.93 56.77 55.64 55.20

201.24 137.98 129.49 71.67 58.58 54.86 56.60 55.46 55.02

207.49 139.07 131.85 72.45 58.86 57.85 56.69 56.04 55.21

212.92 139.85 133.37 73.29 59.08 59.35 56.76 57.50 55.24

215.76 140.26 134.41 73.69 59.20 60.48 56.78 58.67 55.24

216.46 140.47 138.99 73.79 59.25 62.49 56.11 59.75 55.24

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CHEMICAL PHYSICS LETTERS

Volume 168, number 1

Table 4 Electron-correlation energies between the Rydberg electron of HS and the Hd core calculated in different basis sets. For specifications of the CI bases, see text. All energies are in millihartrees State D% 1 ‘E’ 1 2A’

12; 2 lE‘ 2 2A’, 2 2A;’ 3 2E’

1 2E” 3 2A’ I

Orbital basis set (1)98151

(5)95118

(5)95129

(5)95144

(5)95156

(5)95168

(5195

4.02 1.95 2.12 0.54 0.40 0.51 0.40 0.32 0.20

1.70 1.34 1.26 0.23 0.27 0.19 0.31 0.20 0.15

4.20 1.85 2.38 0.52 0.37 0.44 0.37 0.31 0.18

10.45 2.94 4.74 1.30 0.65 1.43 0.46 0.89 0.37

15.88 3.72 6.26 2.14 0.87 2.93 0.53 2.35 0.40

18.72 4.13 7.30 2.54 0.99 4.06 0.55 3.52 0.40

19.42 4.34 11.88 2.64 1.04 6.07 0.54 4.60 0.40

differences between the CI results and corresponding diagonal element of the Hamiltonian matrix are shown in table 4. In fact, the numbers displayed in the table are differences between the correlation

No, of basis

functions

Fig. 2. Rydberg-core correlation energies for the lowest energy levels of H3 versus the number of functions in basis ( 5 ).

energies of the specific state of H3 and the ground state of Hz, calculated using the same one-electron basis set. These numbers are estimates of the correlation between the Rydberg electron and the twoelectron core. The very poor description of the electron correlation in basis ( 1) is evident. It is also observed that the systematic increase of the correlation energy with increasing excitation energy, observed for basis ( 1) breaks down when the bases become larger. However, this regularity remains true within sequences of states of the same symmetry. In general the correlation energy in ‘A; and ‘E” states is larger than in neighbouring ‘A{ and 2E’ states. The Rydberg-core correlation energies for several states are also shown in fig. 2. Their behaviour versus the number of the one-electron basis functions seems to indicate that basis (5) with 95 orbitals is still not saturated and that some of the conclusions concerning the sequence of the energy levels may have to be modified if even larger bases are used. Finally, the calculated and experimental transition energies are compared in table 5. The agreement is good in case of both bases ( 1) and (5 ). Moreover, the agreement remains good if the CI procedure is entirely neglected [ 61 in spite of rather essential differences between the CI eigenvalues and the diagonal matrix elements. This is due to cancellations in calculating the experimental transition energies. Very poor agreement with experiment is obtained in the case of basis (3 ). This result is due to the fact that almost all orbitals in basis ( 1)) 38 orbitals in basis (5 ) but only 23 orbitals in basis (3) are specifically 73

Volume 168, number 1

CHEMICAL PHYSICS LETTERS

20 April 1990

Table 5 Comparison between the experimental and calculated spectra of Hs. Contributions due to electron correlation are given in parentheses. For specification of the CI bases, see text. Transition energies are in millihartrees, except when specified otherwise Transition

1 ‘A,+ 2 ‘E’ 2 ‘A; 1 ‘A;+2’A I 3 =E’ L 3 =A’ 22E’i 22A;I 32E’ 3 ‘A; >

Experimental data ‘)

708-736 556-574 592-615 568-615

nm nm nm nm

Transition energies experimental

(1)98151

(3)86163

(5)95

61.9-64.4 79.4-82.0 74. I-77.0 74.1-80.2

66.4 (1.4) 81.2 (1.4) 71.2 (2.3) 73.0 (2.3) 74.6 (2.5) 13.0 (0.1) 14.9 (0.1) 16.5 (0.3)

71.0 (1.1)

66.7 (1.7) 78.0 ( - 1.7) 79.7 (10.8) 82.2 (11.3) 83.8 (11.5) 14.5 (1.6) 17.0 (2.1) 18.6 (2.2)

3178-3847 cm-’

14.5-17.5

388 l-4456 cm-i

17.7-20.3

3.8 (2.6) 9.7 (2.8) 12.1 (3.6)

‘) Ref. [4].

designed to describe the Rydberg states. In the case of basis (3) the diagonal Hamiltonian matrix elements do not approximate correctly the excited states and the CI expansion is too poor to compensate for this error. These results demonstrate clearly that very large and carefully selected basis sets are necessary in order to obtain an adequate description of both ground and excited (Rydberg) states of a molecule if reliable predictions are required. Certainly, much smaller bases are suflicient to model a (known) Rydberg spectrum. Acknowledgement Partial support to GHFD by Fonds dei Chemischen Industrie im Verband der Chemischen Industrie e.V. is gratefully acknowledged. Travel grants from the Polish Academy of Sciences, contract No. CPBP 0 1.12 and RPBR RRI. 14 are acknowledged by WD and JK. References [I ] G. Herzberg, Ann. Rev. Phys. Chem. 38 ( 1987) 27. [ 21 G. Haberg, J. Chem. Phys. 70 ( 1979) 4806; I. Dabrowski and G. Herzberg, Can. J. Phys. 58 (1980) 1238. [ 31 H. Helm, Phys. Rev. Letters 56 (1986) 42.

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[ 41 M.E. Jacox, J. Phys. Chem. Ref. Data 17 ( 1988) 269.

[ 51 W. Ketterle, Chem. Phys. Letters 160 (1989) 139; W. Ketterle and H. Walther, in: Spectral line shapes, Vol. 5, ed. J. Szudy (Ossolineum, Wro&aw, 1989); A. Dodhy, W. Ketterle, H.P. Messmer and H. Walther, Chem. Phys. Letters 151 (1988) 133; W. Ketterle, H.P. Messmer and H. Walther, Europhys. Letters 8 (1989) 333. [ 61 H.F. King and K. Morokuma, J. Chem. Phys. 71 ( 1979) 3213. [ 71 M. Jungen, J. Chem. Phys. 71 (1979) 3540. [8] R.L. Martin, I. Chem. Phys. 71 (1979) 3541. [ 91 Ch. Nager and M. Jungen, Chem. Phys. 70 (1982) 189. [lo] K.C. Kulander andM.F. Guest, J. Phys. B 12 (1979) L501. [ 111P. Siegbahn and B. Liu, J. Chem. Phys. 68 ( 1978) 2457. [ 121D.G. Trublar and C.J. Horowitz, J. Chem. Phys. 68 (1978) 2466. [ 131B. Liu, J. Chem. Phys. 80 (1984) 581. [ 141M.R.A. Blomberg and B. Liu, J. Chem. Phys. 82 (1985) 1050. [15]A.J.C.Varandas,F.B.Brown,C.A.Mead,D.G.TruhIarand N.C. Blaia, J. Chem. Phys. 86 ( 1987) 6258. [ 161 C.E. Dykstra, A.S. Gaylord, W.D. Gwinn, W.S. Swope and H.F. Schaefer III, J. Chem. Phys. 68 ( 1978) 395 1. [ 171 C.E. Dykatra and W.C. Swope, J. Chem. Phys. 70 (1979)

[ 181i. PreiskomandW. Wotnicki,Mol. Phys. 52 (1984) 1291. [ 191W. Meyer, P. Botschwina and P. Burton, J. Chem. Phys. 84 (1986) 981.

[ 201 C. Urdaneta, A. Largo-Cabrerizo, J. Lievin, G.C. Lie and E. Clemenli, J. Chem. Phys. 88 ( 1988) 209 1, [2 1 ] C.E. Dykstra, Ab initio calculations of the structures and properties of molecules (Elsevier, Amsterdam, 1988).