Relativistic all-electron ab initio calculations for the ground and excited states of the mercury atom

2 September 1994

ELSEVIER

CHEMICAL PHYSICS LETTERS

Chemical Physics letters 227 ( 1994) 229-234

Relativistic all-electron ab initio calculations for the ground and excited states of the mercury atom Matthias M. Gleichmann, Bernd A. He13 Institut fdr Physikalische und Theoretische Chemie, Vniversitiit Bonn, Wegelerstrasse 12, 53115 Bonn, Germany Received 14 June 1994; in final form 28 June 1994

Abstract We present the results of relativistic all-electron ab initio calculations on the mercury atom, using the spin-free no-pair Hamiltonian in a Gaussian basis set expansion. Excitation energies, electron affinity and ionization potential were calculated using a multi-reference configuration interaction technique. The results are in good agreement with experimental data. A new basis set of Gaussian functions for the mercury atom is proposed.

1. Introduction

The experimental study of intermetallic excimers composed of an alkali and a group 12 atom has triggered a number of ab initio calculations on these species [ l-9 1. Compounds involving the heaviest of the transition-metal elements, Hg, require the explicit inclusion of relativistic effects; in recent studies on NaHg [ 1,2] and LiHg [ 10,111 we employed the Douglas-Kroll-transformed no-pair Hamiltonian to implement the relativistic kinematics. In this context, the question about a suitable basis set of Gaussian functions is of importance: should it be optimized in a relativistic procedure? Should it employ a nucleus with finite extension? We have investigated these questions in the particular case of the gold atom [ 12,131 and found that a flexible valence shell is substantially more important than an optimization by means of a relativistic procedure. If, however, a relativistic procedure is used (in our case again the spin-free no-pair Hamiltonian subjected to a Douglas-Kroll transformation ) , then a finite charge distribution for the nucleus is of advantage. A suita-

ble finite-nucleus model prevents the extremely high exponents that arise in a point-nucleus procedure because of the weak singularity developed by the solution of the Dirac equation at the origin. These high exponents are (at best) useless because they describe a region in an atom that is relatively unimportant for the low-order moments of the charge distribution in an atom or a molecule. We therefore proposed to use a basis set optimized in a non-relativistic optimization procedure and then increased its flexibility in the valence region by deleting the small exponents (say those less than 3 bohre2) from the basis set obtained in the optimization procedure, subsequently extending it by means of an even-tempered series or optimizing the outer valence region by means of configuration interaction calculations. In this Letter, we employ this procedure for the development of a Gaussian basis set for the mercury atom starting from the non-relativistic (unpublished) basis set that was used for our previous NaHg work [ 1,2 1. We assess the flexibility of our basis set for the calculation of the spectroscopy of the inter-

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metallic excimers by all-electron configuration interaction (Cl) calculations of the low-lying electronic spectrum of the Hg atom, which in turn is compared with experiment, quasirelativistic effective core potential (ECP) contracted Cl (Ccl) calculations [ 141, recent quasirelativistic pseudopotential configuration interaction calculations [ 15,161, and numerical relativistic calculations by Migdalek et al. [ 17-201.

2. Details of the calculations 2.1. Theoretical method The Hamiltonian used in our relativistic calculations is the spin-free no-pair Hamiltonian H$’ with external field projectors correct to second order in the external potential. H$’ is obtained through a Douglas-Kroll transformation of the no-pair Hamiltonian [ 2 l-241 correct to second order in the external potential. Since this Hamiltonian is well-documented in the literature, we refrain from giving any details here, and just mention the most important approximations made in the present framework: the unmodified non-relativistic Coulomb potential for the electron-electron interaction is used; the operator averages over tine-structure components, so one obtains a one-component formalism. Adopting these approximations, only a modification of the one-electron integral evaluation is required. H”,” can therefore be easily implemented in common ab initio program packages. In particular, we employ complete active space self-consistent field (CASSCF) [ 25 ] and Cl methods to calculate the excited states of mercury, thus treating static and dynamic effects of electron correlation. 2.2. Basis set As a starting point, we used the basis set employed in our study on NaHg [ 1,2 1, which was obtained by straightforward optimization of the Hg ground state in a nonrelativistic SCF procedure 126-281 and comprised 73 Hg basis functions. In order to increase the flexibility of the outer valence, we deleted the smallest exponents and in turn augmented the basis set by 4 s, 3 p, 2 d and 3 f primitives according to an even-tempered series. For an explicit representation

of one-particle functions for the excited ‘P and ‘P states we optimized the three most diffuse p functions by minimizing the total energy of these states at the Cl level. The molecular orbital coefficients of a relativistic SCF calculation with a totally uncontracted basis set were used to determine ,the contraction coefficients of the s, p and f orbitals. The d primitives remained uncontracted at this stage. The contraction coefficients of the d orbitals were subsequently obtained in the following way: a relativistic CASSCF calculation for the ground state, applying the spin-free no-pair operator, was the starting point. Two active electrons (the 6s electrons) were allowed to occupy four active orbitals (6s, 6p, 6p,,, 6p,) in order to describe the near-degeneracy of the 6s6p configuration. The next step was an ACPF (averaged coupled-pair functional) [29] calculation, using the four canonical reference configurations for the correlation of two electrons. The ‘natural orbitals’ obtained from a diagonalization of the ACPF one-particle density matrix were employed to contract the d shells. The final basis set comprises 97 functions and is shown in Table 1.

2.3. The CI calculations

The CASSCF orbitals taken from the procedure described above were used as many-particle basis of our Cl calculations, which were performed with the multireference single and double excitation method (MRD-Cl) [ 301. All energies were extrapolated to estimated full-cl values by use of the generalized Davidson formula [ 3 11. A selection threshold of 10 pH was used to keep the dimension of the explicitly solved secular equation small. The number of generated and selected configurations is shown in Table 2. Anticipating the requirements for the treatment of excimers MeHg, Me = alkali metal, we were mainly interested in the Hg states ‘S, ‘S, ‘P, and ‘P with closed d shells. In order to assess the importance of the 5p core polarization, test calculations with 18 correlated electrons were performed which showed that there was no significant difference ( < 0.03 eV) to the 12 electron Cl treatment, which was employed in the following.

Table 1 Basis set of the mercury atom, employing a generalized (2 1s 17p 1Od7f ) + Function

[ 14s 11p5d2f] contraction of Gaussian functions

Exponent

Relativistic contraction coefficients

3260982 469370.5 105124.2 29609.54 9741.310 3603.125 1467.629 639.1015 251.7650 126.8379 67.05787 28.32247 14.52412 5.496432 2.200000 0.880000 0.350000 0.140000 0.056000 0.022000 0.010000

0.007479 0.018744 0.044032 0.093634 0.184200 0.304539 0.349520 0.197130 0.026865 -0.005548 0.002537 -0.001174 0.000626 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.002924 0.007405 0.017746 0.038905 0.082102 0.149271 0.214213 0.0606 11 -0.457275 -0.487974 -0.191534 -0.001128 -0.003498

0.001404 0.003559 0.008568 0.018829 0.040458 0.073899 0.113835 0.017307 -0.356495 -0.632245 0.163508 0.989209 0.173328

0.000704 0.001785 0.004299 0.009470 0.020357 0.037557 0.057724 0.009 15 1 -0.211298 -0.379111 0.119353 1.357389 -0.385309

25023.70 6000.760 1956.519 746.2659 312.4783 138.6019 59.46798 27.96633 11.54251 5.305651 2.400000 1.100000 0.500000 0.230000 0.100000 0.042000 0.024000

-0.007466 -0.025802 -0.088428 -0.233671 -0.413899 -0.352783 -0.083782 0.005862 -0.003458 0.002396 1.0 1.0 1.0 1.0 1.0 1.0 1.0

0.003903 0.013644 0.048382 0.132415 0.254263 0.133969 -0.457997 -0.581597 -0.111282 0.014339

-0.001951 -0.006858 -0.024390 -0.067893 -0.131305 -0.058674 0.378290 0.418107 -0.577248 -0.614239

- 0.00079 1 -0.002785 -0.009905 -0.027703 -0.053578 -0.023084 0.171371 0.180241 -0.376467 -0.430554

Id 2d 3d 4d 5d 6d 7d 8d 9d 10d

861.0782 255.9001 95.16132 38.74511 14.82950 5.923862 2.400000 1.oooooo 0.400000 0.160000

0.026094 0.151009 0.419946 0.884874 0.138252 -0.010811 0.007753 -0.004255 0.002195 -0.001052

-0.013847 -0.080936 -0.232190 -0.178300 0.460550 0.592611 0.095968 0.010678 -0.004677 0.002478

0.004305 0.025168 0.072804 0.048960 -0.184495 -0.205177 0.258632 0.504608 0.366879 0.113618

-0.005116 -0.029744 -0.086918 -0.054639 0.224810 0.269243 -0.568953 -0.494899 0.668349 0.393631

If 2f 3f 4f

91.47338 29.27513 10.50179 3.569958

0.068840 0.298252 0.533547 0.374704

5f 6f 7f

1.200000 0.460000 0.200000

0.657025 0.423457 0.093790

IS

2s 3s 4s 5s 6s 7s 8S 9s 10s 11s 12s 13s 14s 15s 16s 17s 18s 19s 20s 21s 1P 2P 3P 4P 5P 6~ 7P 8~ 9P 1OP 1lP 12P 13P 14P 15P 16~ 17P

0.000308 0.000782 0.001883 0.004147 0.008928 0.016464 0.025454 0.003747 -0.095116 -0.177354 0.064261 0.744222 -0.283281

0.005518 0.031238 0.095313 0.046380 -0.219666 -0.416093 1.322435 -0.845321 -0.578635 0.983868

0.000089 0.000226 0.000545 0.001201 0.002587 0.004770 0.007380 0.001075 -0.027694 -0.051922 0.019205 0.224341 -0.089311

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Table 2 Results and details of the MRD-CI calculations on the Hg atom. Energies ’ in eV E =wd

State

CSFs gen. b

CSFs sel. b

Configuration

nMmR ’

No.R

LCf

‘S 3P ‘P ‘S ‘S ‘P

113715 257311 163961 408767

7386 9345 7600 8797

5d’06s2 5d”6s16p’ 5d1’6s’6p’ 5d1’6s’7s’ 5d”6s17s’

34M4R 34M4R 39M4R 42M4R 34M4R

l.R l.R l.R l.R 2.R

0.94 0.95 0.94 0.95 0.95

0.00 5.20 6.75 1.52 7.70

5d96s26p’ 5d”6s’7p1

e

“P ‘P ‘D ‘D ‘D ?S ‘S

5d”6s’7p1 5d1’6s’6d’ 5d”6s’6d1 5d96s26p’ 5d1’6s’8s’ 5d’06s’8s’

34M4R 39M4R f f

2.R 2.R

0.94 0.94

8.82 9.14

:2M4R 34M4R

2.R 3.R

0.95 0.94

9.28 9.39

8.74 8.84 8.84 8.85 9.07 9.17 9.22

10.17

10.44

IE EA

EEFCI

0.00 5.18 6.70 7.73 7.86

8.54

17175

3833

5d”6s’

4M2R

l.R

0.95

220435

9944

5d”6s26p’

27M4R

l.R

0.94

-0.88

g

’ ‘Estimated full-CI’ energies [ 3 11. b For each symmetry, the number of generated and selected configuration state function (CSFs) is given. Higher states of same symmetry are obtained as higher roots in the generated space. c A configuration space generated from n reference configurations was used for selection with respect to m roots of the

zero-ordersecular

equation. dDatafrom [ 321. c Stateswithclosedd shellwereconsideredonly. ‘The basis set did not comprise Hg6d Rydberg functions.

* No experimental values available.

3. Results and discussion Our results for the low-lying excited states of the mercury atom are given in Table 2. It is well known [ 171 that both relativistic and correlation effects are important for a realistic determination of the electronic spectrum of the Hg atom. To illustrate this, we mention the poor values for the ionization potential at non-relativistic Hartree-Fock level (6.83 eV) or Dirac-Hartree-Fock level (8.55 eV) in comparison to experiment ( 10.44 eV). Migdalek and Baylis investigated excitation energies and oscillator strengths to the low-lying %’ and ‘P states as well as the first ionization potentials of the mercury isoelectronic series [ 18 ] by means of numerical multi-configuration relativistic Hartree-Fock (MCRHF) and relativistic configuration interaction based on Hat-tree-Fock orbitals (CIRHF) [ 201. In these studies, intermediate coupling is achieved by means of a limited (threeconfiguration) multiconfiguration Hartree-Fock procedure or a corresponding configuration interaction method. The effect of core-valence correlation

is recovered by means of a core polarization potential, and spin-orbit coupling is automatically included because the Hamiltonian is based on the fourcomponent Dirac formalism. The results obtained are excellent: the excitation energies for the ‘So-‘Pi and ‘So-3PI transitions agree with experiment to within 0.05 eV. Unfortunately, these results cannot be compared with ours directly, since the values for the other fine-structure components are not given and thus a spin-averaged value cannot be deduced. The ionization potential in the MCRHF ( 10.00 eV) and the CIRHF approach (9.65 eV) shows a larger deviation from experiment ( 10.44 eV), presumably because of limitations of the core-polarization model. Our result ( 10.17 eV) indicates that a suitable inclusion of dynamic correlation is able to bring the theoretical results into better agreement with experiment. Moreover, our experience with 18 correlated electrons shows that the large core-polarization effect found by Migdalek and Baylis is mostly due to the 5d shell: explicit correlation results in excitation energies in substantial agreement with experiment.

M.M. Gleichmann,B.A. He8 /Chemical PhysicsLetters227 (1994) 229-234

233

Table3 Comparison of our spin-orbit averaged excitation and ionization energies (eV) of Hg with recent quasirelativistic pseudopotential configuration interaction calculations and experimental data State

‘S

‘P ‘P ?S

Configuration

5di06s2 5d’06s’6p’ 5di06s’6p’ 5d”6s’

QR-ECP

CISD

PP

CASSCF

cc1 ]141

r151

CASSCF + MRCI ]151

+ ACPF ]15]

AD CASSCF + MRDCI (this work)

Exp. [34

0.00 4.86 6.51 10.04

0.00 5.22 6.90 10.20

0.00 5.23 6.91 10.25

0.00 5.20 6.15 10.17

0.00 5.18 6.70 10.44

0.00 5.13 10.14

Theoretical excitation energies for the higher excited states as given in Table 2 have not been reported. We note in this context that in an LCAO approach (a linear combination of atomic orbitals) it is most important to have an optimized outer function for the excited electrons: the ‘P and ‘P states, for each of which a specific p exponent has been optimized, display the best agreement; higher excited states for which no specifically optimized function is available, show less accurate results, e.g., the second excited states of ‘P or 3P symmetry, or the ‘D state, which could not be obtained at all because a corresponding 6d Rydberg function has not been included in the basis set. Likewise, a treatment resulting in satisfactory accuracy for the states with an open d shell requires more effort. Since the design of the basis set aimed at the specific purpose mentioned in Section 1, we did not attempt to optimize the basis set to enable the calculation of states with an open d shell. The electron affinity of the Hg atom is definitely negative. Our ionization potential and the spin-averaged excitation energies may be directly compared with quasirelativistic ECP results [ 141 and quasirelativistic pseudopotential results [ 15 ] employing the Wood-Boring operator [ 331, see Table 3. The results obtained are in good agreement with ours. The authors have already stressed the importance of multireference effects, which may be clearly deduced from the relatively poor results of a single-reference singles-and-doubles CI (CISD ) . 4. Conclusion

A new Cartesian Gaussian-type orbital basis set for the mercury atom has been presented. It has been used

in relativistic CASSCF and MRD-CI calculations, applying the spin-free Douglas-Kroll transformed nopair Hamiltonian with external field projectors correct to second order in the external potential in a basis set expansion. Spin-averaged excitation energies are in excellent agreement with experiment and quasirelativistic pseudopotential CI calculations.

Acknowledgement The present work has been funded by the Deutsche Forschungsgemeinschaft in the framework of the ‘Schwerpunkt Atom- und Molekilltheorie’. Support from the European Science Foundation and the Fonds der Chemischen Industrie are gratefully acknowledged.

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