Time-dependent DFT study on electronic states of vanadium and molybdenum oxide molecules

18 May 2001

Chemical Physics Letters 339 (2001) 433±437

www.elsevier.nl/locate/cplett

Time-dependent DFT study on electronic states of vanadium and molybdenum oxide molecules Ewa Brocøawik a,*, Tomasz Borowski b a

Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Niezapominajek 8, 30 239 Cracow, Poland b Faculty of Chemistry, Jagiellonian University, Ingardena 3, Cracow 30-060, Poland Received 20 December 2000; in ®nal form 20 March 2001

Abstract Spin-unrestricted time-dependent density functional theory (TDDFT) calculations for excited states of VO and MoO molecules have been undertaken to validate its applicability to highly open-shell systems. Equilibrium geometries, vibrational frequencies and excitation energies are compared with experimental data and DSCF DFT calculations where available. Overall good performance of TDDFT for intricate spectroscopic properties of transition metal (TM) oxides is found. Examples where discrepancies between experiment and theory could be expected are spotted and discussed. Ó 2001 Published by Elsevier Science B.V.

1. Introduction Time-dependent DFT methodology [1,2] has proved to be an inexpensive and reliable method for calculating excitation energies for a wide range of organic [3±6] and inorganic [7,8] molecular systems. However, most of the TDDFT results reported up to date concern closed-shell organic or inorganic molecular systems. Only very recently a few publications considering applicability of TDDFT for open-shell small organic molecules have been published [9±11]. As to the best of our knowledge there are no TDDFT results reported for transition metal (TM) openshell systems. We have performed calculations for two TM oxides, namely VO and MoO molecules. These molecules are good testing systems because

*

Corresponding author. Fax: +48-12-634-05-15. E-mail address: [email protected] (E. Brocøawik).

of thorough spectroscopic characterisation available [12,13] and the well-understood electronic structure with single-determinantal ground state function [14,15]. 2. Methodology The ground state geometry of VO and MoO was optimised within unrestricted Kohn±Sham formalism employing a three-parameter B3LYP functional for exchange and correlation (xc) [16]. Four di€erent basis sets were tested for VO. Three of them were all-electron basis sets from 6-311G family, namely 6-311G(d), 6-311+G(d) and 6-311+G(2d). They consisted of Wachters and Hay's [17,18] basis on vanadium atom and Pople's basis on oxygen [19]. This particular choice allowed us to assess the importance of the di€use and polarisation functions for the molecular excitation energies. The fourth basis set was

0009-2614/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 0 3 6 1 - X

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LANL2DZ ± valence double zeta basis set with e€ective core potential (ECP) replacing 1s, 2s, 2p core electrons on V atom [20,21] and DZ basis set on oxygen [22]. For MoO we have restricted our calculations to the LANL2DZ basis set, because we have found this basis to perform very well for VO and because it contains relativistic pseudopotential for Mo which should be signi®cant for the second row TMs. Time-dependent calculations employed the same B3LYP xc functional as the SCF step (adiabatic approximation). The excited states potential energy curves were constructed by scanning the bond distance and starting TD computations from the converged ground state density for the given bond length. These results were subsequently used as input for the program LE V E L [23] to solve the nuclear Schr odinger equation and to evaluate vibrational frequencies and bond equilibrium distances. Adiabatic excitation energies (D) were then calculated as the di€erences between ®tted energies at potential minima. CASSCF(10,10) calculations were also performed for inspection of the ground state wavefunction with the use of MO L C A S 5 program [24]. Also limited CASPT2 calculations were undertaken for MoO molecule to compare the performance of the two correlated methods. LANL2DZ basis set common for the two oxides has been employed. The SCF and TDDFT [25] computations were performed with the use of A.9 version of GA U S S I A N 98 [26] suite of programs. 3. Results and discussion In Table 1 we give the dominant con®gurations for the electronic states of VO and MoO studied here. CASSCF calculations strongly supported single-determinantal character of the ground state wavefunction for both oxides for full range of bond distances used in TDDFT calculations  The weight of the dominant con(Re  0:15 A). ®guration (listed in Table 1) in CASSCF wavefunction fell between 0.75 and 0.90. Another important aspect is that all of the selected quartet states for VO and septets for MoO, and all but one

Table 1 Dominant con®gurations and spectroscopic states of the VO and MoO molecules State

Con®guration

VO X4 R 4

r2b p4b r14s d23d 04

A P; A U

r2b p4b r14s d13d p13d

4

BP

r2b p4b d23d p13d

C4 R

r2b p4b d23d r1 3d

4

DD

r2b p4b r14s d13d r1 3d

MoO X5 P

r2b p4b d24d r15s p14d

A5 R‡

r2b p4b d24d p24d

5

r2b p4b d24d r15s r14d

R 05

r2b p4b d14d r15s p24d

5

r2b p4b d24d p14d r14d

05

B P; 5 U

r2b p4b d14d r15s p14d r14d

7

r2b r3b d24d r15s p24d

A D BP

7

P ‡

R

r1b p4b d24d r15s p24d

quintets for MoO may be derived from the ground state by a single-electron promotion. Only B0 5 P, 5 U states of MoO in real representation for degenerate p and d orbitals have the wavefunctions consisting of two determinants, one of which differs from the ground state by two excitations. As the TDDFT scheme used here allows only singleelectron promotions, these states of MoO cannot by described by this methodology in a reliable fashion. Indeed, numerical results support this prediction (vide infra). In Table 2, we present our results for quartet states of VO. They are compared with experiment [12,13] and previous DSCF DFT calculations within LDA approximation [14,15]. Several systematic observations can be made on the basis of these results. First of all, an inclusion of the di€use functions improves the results dramatically in allelectron computations. The most striking di€erences are found for the excitations to the B4 P and C4 R states calculated with 6-311G(d) and 6-311+G(d) basis sets. This may be explained on the basis of the electron con®gurations these states stem from. These are the only states of VO studied here with empty r4s orbital from which electron was promoted to either p3d or r3d . Thus, these

E. Brocøawik, T. Borowski / Chemical Physics Letters 339 (2001) 433±437

435

Table 2 DFT/TDDFT results for the VO molecule State

Exp.a

DSCF DFTb

TDDFTc 6-311G(d)

 Re (A) X4 R A0 4 U A4 P B4 P C4 R D4 D

1.592 1.629 1.637 1.644 1.675 1.686

1.58

D …cm 1 † X4 R A0 4 U A4 P B4 P C4 R D4 D

±

7255 9899 12 606 17 420 19 148

Mean abs. error x …cm 1 † X4 R A0 4 U A4 P B4 P C4 R D4 D Mean abs. error

1002 936 884 901 852 835

6-311+G(2d)

LANL2DZ

1.63 1.65 1.66

1.582 1.623 1.637 1.653 1.701 1.675

1.584 1.625 1.617 1.647 1.660 1.659

1.586 1.626 1.619 1.648 1.660 1.659

1.611 1.650 1.644 1.665 1.677 1.683

0.02

0.010

0.013

0.012

0.012

1.62

Mean abs. error

6-311+G(d)

±

±

±

±

±

11 971 17 340 20 958

8581 13 711 17 504 28 428 22 428

5847 11 085 10 295 18 121 17 309

5886 11 140 10 245 18 009 17 336

6114 11 982 9297 17 767 18 221

820

4865

1489

1474

1561

993 935 965 958

1062 1002 963 969 672 684

1029 964 960 937 900 879

1028 965 969 935 920 884

1003 953 943 942 934 799

25

101

43

49

39

9328

956

a

Ref. [12]. b Ref. [14]. c This work.

states originate from the di€erent atomic con®gurations than the other quartet states. It is well documented that the spatial extent of the 3d orbitals in ®rst row TMs increases dramatically in the series: 3dn 2 4s2 < 3dn 1 4s1 < 3dn , and that a well-balanced description of the 3d orbitals is needed to reproduce the energy di€erences in these series [18]. Thus the di€use functions in the 6-311+G basis set seem to be indispensable for the reliable description of these excited states. Secondly, the e€ect of the additional set of polarisation functions seems to be not so important. The overall good performance of the LANL2DZ basis set should be stressed here. This modest basis set gives results of quality comparable with the biggest all-electron basis set employed.

Bond distances and vibrational frequencies for the ground and excited states are found to be within the typical error of the B3LYP SCF calculations for all basis sets tested here. As regards the adiabatic excitation energies, none of the basis sets reproduced the proper ordering of all states. For example A4 P and B4 P states are interchanged by 6-311+G(d), 6-311+G(2d), and LANL2DZ. This disagreement originates probably from the well-known problem in well-balanced description of the atomic states in vanadium coming from di€erent 3d4 x 4s1‡x con®gurations. Only 6-311G(d) basis set gave these states in proper order, but the absolute energies were much too high. Any of the all-electron basis sets gave the inverse ordering of the highest two

436

E. Brocøawik, T. Borowski / Chemical Physics Letters 339 (2001) 433±437

states, namely C4 R and 4 D. LANL2DZ gave them in the correct order, although their spacing was too small. This problem may be addressed to the description of r3d orbital occupied in both states. This high-lying orbital appeared unbound in the smallest all-electron basis set (where dramatic disagreement with experiment has been found) while it became bound for other bases. For all tested properties we have calculated mean absolute errors from experiment. Apart from the smallest all-electron basis set where this error is large, other basis sets gave errors of comparable magnitude. They are in agreement with the TDDFT/B3LYP results for other molecular systems. On the basis of this comparison, the LANL2DZ basis set may be advertised as the modest basis with acceptable accuracy. Thus this basis has been selected for calculations for molybdenum oxide where the use of ECP is highly desirable. The results for MoO given in Table 3 are very encouraging. Here the excitation energy errors do not exceed 1000 cm 1 for the ®rst four excited states. The last quintet state B0 5 P is one component of the multiplet derived from the d4d r5s p4d r4d con®guration which cannot be accessed from the MoO ground state by single excitation. Thus this state is beyond the reach of this TDDFT scheme and such a big discrepancy could be expected here.

Table 3 DFT/TDDFT results for the MoO molecule  Re (A) State a

5

Pr A5 R‡ 5 R A0 5 D B5 P B0 5 P; 5 U 7 P 7 ‡ R Mean abs. error a

The results presented here support the TDDFT methodology as a reliable tool for calculating excitation energies for the TM systems with many open-shell electrons. It provides a natural way of accessing multiplet problems and higher excited states within a common symmetry manifold. This may be illustrated by fA4 P; A0 4 Ug states of VO where only a mean excitation energy could be estimated from DSCF DFT calculations or the B5 P state of MoO which was unreachable in that methodology. However, one should be aware of its limitations and care should be taken of the nature of excited states in question. Some states may be not accessible, as they di€er from the ground state by more than one excitation within a given manifold (alpha/beta). Some open-shell ground states may have intrinsic multideterminantal character due to spin couplings where not all components are accessible in a straightforward way. Nevertheless, the high cost e€ectiveness of TDDFT calculation scheme (the time ratio of CASPT2 versus TDDFT for single-point calculations equalled here to 150) puts this methodology in the category of reliable and very ecient tools for correlated electronic structure calculations.

D …cm 1 † c

x …cm 1 †

Exp.

DSCF DFTb

TDDFT

Exp.

DSCF DFT

TDDFT

Exp.

DSCF DFT

TDDFT

1.70 ± ± 1.75 1.76 1.73 ± ±

1.73 1.77 1.765 1.78 ± 1.80 2.03 2.00

1.750 1.782 1.779 1.776 1.823 1.849 2.038 1.989

±

±

±

893 ± ± 867 ± ± ± ±

879 920 941 944 ± 941 550 580

902 962 854 990 779 636 565 499

0.04

0.07

51

66

Ref. [13]. Ref. [15]. c This work, LANL2DZ basis set. b

4. Conclusion

7350 13 373 14 362 20 723 21 712 ± ±

7450 12 420 13 510 ± 29 370 24 750 31 850

7829 12 551 14 966 20 582 27 169 16 297 24 497

2390

1501

E. Brocøawik, T. Borowski / Chemical Physics Letters 339 (2001) 433±437

Acknowledgements This study was sponsored by the Polish State Committee for Scienti®c Research (Grant no. 3 T09A 130 19). The computing facilities were supported by a grant from the State Committee for Scienti®c Research (KBN/SGI_ORIGIN_2000/ UJ/042/1999). References [1] E.K.U. Gross, C.A. Ullrich, U.J. Grossmann, in: E.K.U. Gross, R.M. Dreizler (Ed.), Density Functional Theory, Plenum Press, New York, 1995, p. 149. [2] M.E. Casida, in: J.M. Seminario (Ed.), Recent Developments and Applications of Modern Density Functional Theory, Elsevier, Amsterdam, 1996, p. 391. [3] R. Bauernschmitt, R. Ahlrichs, Chem. Phys. Lett. 256 (1996) 454. [4] K.B. Wiberg, R.E. Stratmann, M.J. Frisch, Chem. Phys. Lett. 297 (1998) 60. [5] S. Hirata, T.J. Lee, M. Head-Gordon, J. Chem. Phys. 111 (1999) 8904. [6] S.J.A. van Gisbergen, A. Rosa, G. Ricciardi, E.J. Baerends, J. Chem. Phys. 111 (1999) 2499. [7] S.J.A. van Gisbergen, J.A. Groeneveld, A. Rosa, J.G. Snijders, E.J. Baerends, J. Phys. Chem. A 103 (1999) 6835. [8] A. Rosa, E.J. Baerends, S.J.A. van Gisbergen, E. van Lenthe, J.A. Groeneveld, J.G. Snijders, J. Am. Chem. Soc. 121 (1999) 10356.

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