Scholarly article on topic 'Excited states of the water molecule: Analysis of the valence and Rydberg character'

Excited states of the water molecule: Analysis of the valence and Rydberg character Academic research paper on "Physical sciences"

0
0
Share paper
Academic journal
The Journal of Chemical Physics
OECD Field of science
Keywords
{""}

Academic research paper on topic "Excited states of the water molecule: Analysis of the valence and Rydberg character"

Excited states of the water molecule: Analysis of the valence and Rydberg character

Mercedes Rubio, Luis Serrano-Andrés, and Manuela Merchán

Citation: The Journal of Chemical Physics 128, 104305 (2008); doi: 10.1063/1.2837827 View online: http://dx.doi.org/10.1063/1-2837827

View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/128/10?ver=pdfcov Published by the AIP Publishing

Articles you may be interested in

Theoretical description of electronically excited vinylidene up to 10 eV: First high level ab initio study of singlet valence and Rydberg states

J. Chem. Phys. 141, 174317 (2014); 10.1063/1.4900875

Adiabatic versus diabatic descriptions of the lowest Rydberg and valence I + 1 states of HCl J. Chem. Phys. 134, 204104 (2011); 10.1063/1.3590360

Orbital optimization in the density matrix renormalization group, with applications to polyenes and p -carotene J. Chem. Phys. 128, 144117 (2008); 10.1063/1.2883976

Computational studies on the ground and excited states of BrOOBr J. Chem. Phys. 124, 204309 (2006); 10.1063/1.2197831

Multireference configuration interaction study of the mixed Valence-Rydberg character of the C 2 H 4 1 (n,n * )V state

J. Chem. Phys. 106, 7208 (1997); 10.1063/1.473275

Excited states of the water molecule: Analysis of the valence and Rydberg character

Mercedes Rubio,1,2,a) Luis Serrano-Andrés,1 and Manuela Merchán1

1Instituto de Ciencia Molecular, Universitat de Valencia, Apartado 22085, ES-46071 Valencia, Spain 2Fundació General de la Universitat de Valencia, Plaza del Patriarca, 4, 1°, ES-46002 Valencia, Spain

(Received 8 November 2007; accepted 3 January 2008; published online 12 March 2008)

The excited states of the water molecule have been analyzed by using the extended quantum-chemical multistate CASPT2 method, namely, MS-CASPT2, in conjunction with large one-electron basis sets of atomic natural orbital type. The study includes 13 singlet and triplet excited states, both valence and 3s-, 3p-, and 3d-members of the Rydberg series converging to the lowest ionization potential and the 3s- and 3p-Rydberg members converging to the second low-lying state of the cation, 1 2A1. The research has been focused on the analysis of the valence or Rydberg character of the low-lying states. The computation of the 1 1B1 state of water at different geometries indicates that it has a predominant 3s-Rydberg character at the equilibrium geometry of the molecule but it becomes progressively a valence state described mainly by the one-electron 1b 1 ^ 4a1 promotion, as expected from a textbook of general chemistry, upon elongation of the O-H bonds. The described valence-Rydberg mixing is established to be originated by a molecular orbital (MO) Rydbergization process, as suggested earlier by R. S. Mulliken [Acc. Chem. Res. 9, 7 (1976)]. The same phenomenon occurs also for the 1 1A2 state whereas a more complex behavior has been determined for the 2 1A1 state, where both MO Rydbergization and configurational mixing take place. Similar conclusions have been obtained for the triplet states of the molecule. © 2008 American Institute of Physics. [DOI: 10.1063/1.2837827]

I. INTRODUCTION

Although the spectroscopic properties of the water mol-

ecule have been studied since long time ago they are still

subject of current interest, greatly motivated by the chemical and biological importance of the system. In particular, the characterization of the low-lying electronic excited states as valence or Rydberg is an open debate which requires a careful theoretical analysis from the viewpoint of modern ab initio quantum chemistry. A number of earlier studies reported the absorption and photoionization cross sections of water vapor in the vacuum ultraviolet region,2,9-13 (see also Ref. 14 for a review of the earlier work), as well as excitation energies obtained by electron-impact spectroscopy.15,16 The observed bands were interpreted as Rydberg series on the basis of the quantum defect analysis and using available ab initio calculations as a guide for the assignments.11-13,16-19 A comprehensive ab initio study on the excited states of

water performed in 1974 by Goddard and Hunt characterized all the computed states below 11.7 eV as having Rydberg nature, a result supported at the configuration interaction (CI) level.18,19 As shall be discussed below, the electronic spectrum of water in the gas phase is currently interpreted as composed either of different Rydberg series or implying excited states with a significant Rydberg character at the ground-state equilibrium geometry (see, for instance, Ref. 20 for a more recent review). How can this scenario

Author to whom correspondence should be addressed. Electronic mail: mercedes.rubio@uv.es.

conciliate with our simple knowledge of the electronic structure of the water molecule? It is obvious that the ground state of the system can be described as having two valence unoccupied antibonding molecular orbitals (4a1 and 2b2 in C2v symmetry), and corresponding valence excited states could be expected. The present study attempts to throw more light into this aspect of the excited states of water which we consider has been somewhat overlooked.11,14,21-23

Despite earlier conclusions which emphasized the Ryd-berg character of the band,17-19 the experimental evidence does suggest some valence character for the lowest-energy band of the gas-phase electronic spectrum of water.11,14,23 The band is broad (6.8-8.2 eV), with poorly defined vibra-tional progressions, and has its maximum absorption around 7.4 eV.4 These features result from direct photodissociation of water into ground-state H- and OH- radicals, as shown experimentally24,25 and theoretically.26 There is a unanimous agreement in assigning the lowest-energy band to the 1 1 B1 state, usually named A 1B1 state according to Herzberg's

notation, but not in the valence, Rydberg, or mixed valence-Rydberg nature of the state. As far as we know, only Wang et al.1 claimed a significant valence character, ascribed to the one-electron 1b1 ^ 4a1 promotion, for the band at 7.4 eV and proposed a different location for the related 1B1 (1b 1 ^ 3s) Rydberg state. In contrast, according to Robin14 and using his own words: "the valence 1b1 ^ 4a1 configuration is totally dissolved in the conjugated Rydberg sea," and therefore none of the excited states of water could be considered as having valence nature. The lowest-energy

0021-9606/2008/128(10)/104305/12/$23.00

128, 104305-1

© 2008 American Institute of Physics

(n = 3) member of the ibi ^ ns Rydberg series was, however, predicted to have the largest contribution of the 4ai valence orbital22 in line with experimental evidence.11,14 A somewhat different point of view (see Ref. 14 for discussion) was provided by Mulliken, who described the i iBi state of water as a semi-Rydberg or near-Rydberg state meaning that there is only one state described by a valence-Rydberg mixing in the

wave function, a character which is in addition affected by

the internuclear distance. Moreover, Mulliken explained the change in the nature of the state with the internuclear distance as a result of the Rydbergization of the molecular orbital (MO) where the electron is excited to; namely, the orbital corresponds to a 3s-Rydberg MO at short O-H distances and close to the ground-state equilibrium geometry, increasing its valence character (4ai) when elongating the O-H bond. To our knowledge, the composition of the 3s/4ai orbital as a function of the O-H distance was analyzed only

in an earlier study performed by Horsley and Fink. Chergui

and Schwentner also attempted to show experimentally the MO Rydbergization postulated by Mulliken for the i iBi of water. On the other hand, on the basis of the different shape of the bands recorded when studying photodissociation chan-

nels of water, Dutuit et al. proposed a valence character for the nsai-Rydberg series. Furthermore, they attributed the valence character of the 3sai-Rydberg states to MO Rydbergization as described by Mulliken.21,28

In the last decade, different theoretical studies on the excited states of water have been carried out using coupled-cluster methods,6,31 density functional theory (DFT) approaches,5,32 and multireference CI techniques.33 Some of the studies31,34 relate the contribution of the antibonding valence 4ai orbital to the lowest singlet excited state, A lB,, to

the dissociative nature of the state. However, the character of the low-lying excited states as function of the O-H internuclear distance has not been analyzed and theoretical efforts32,33 have been mainly focused on determining the low-lying potential energy surfaces of water related to photodissociation processes of the molecule. We report here a high-level ab initio study of the singlet and triplet excited states of water by using multiconfigurational second-order perturbation theory through the CASPT2 method.35,36 This methodology is nowadays a well-established approach to accurately compute excited states in different types of compounds (see, for instance, Refs. 37-40). Furthermore, the CASPT2 method is particularly suited for the characterization of potential energy surfaces because of the inherent flexibility of the complete active space self-consistent field (CASSCF) wave function used as reference in the second-order perturbation treatment.4i To the best of our knowledge, only results for the i iBi and i 3Bi states have been previously reported at the CASPT2 level, but the analysis was

mainly focused on their electric properties. The present study provides accurate excitation energies for the singlet and triplet excited states of the water molecule at the ground-state structure and at several geometries along the symmetrical stretching dissociation channel, paying special attention to the nature of the excited states. In the light of these results, we expect to achieve a better understanding of the Rydberg/ valence character of the electronic spectrum of gas-phase

water. Such knowledge is crucial for rationalizing the observed blueshift of the absorption spectrum of liquid water and ice since Rydberg and valence states are differently affected in condensed phases.14

II. METHODS AND CALCULATION DETAILS

In order to study the Franck-Condon (FC) region of the molecule, calculations were performed at the experimental equilibrium geometry of the ground state reported by Herzberg (rO-H = 0.958 A and <HOH= 104.5°, C2v symmetry).27 According to the standard orientation, the molecule was placed in the yz plane with the z axis corresponding to the C2 symmetry axis. In order to analyze the character of the electronic states at different molecular geometries, potential energy curves obtained by symmetrically stretching the O-H bond lengths have also been computed. Generally contracted basis sets of the atomic natural orbital (ANO-L) type were used, which were built from the O(14s9p4d3f)/H(8s4p3d) primitive sets.43 These basis sets were designed to optimally treat correlation and polarization effects. In order to select the contraction scheme to be used, the vertical excitation energy to the 1 1B1 state was computed at the CASPT2 level using the full valence active space and several basis sets. The results are collected in Table I together with the values of the second moments of the electronic charge distribution computed at the CASSCF level. To properly describe Rydberg states, the ANO-L type basis sets were supplemented with two s (exponents 0.023 48 and 0.008 22), two p (exponents 0.015 79 and 0.005 53), and one or two d (exponents 0.056 27 and 0.019 70) diffuse functions placed at the oxygen atom position. The exponents of the diffuse functions were determined by an even-tempered expansion from the primitive sets. As shown in Table I, the ANO-L O[5s4p1d] / H[2s1p] contraction (basically a quadruple- and double-zeta plus polarization scheme for oxygen and hydrogen, respectively) yields a transition energy for the 1 1B1 state differing ±0.02 eV from that obtained with larger basis sets, including up to f functions, and more flexible contraction schemes. The addition of diffuse functions has also a small effect on the computed excitation energy and second moments. It is worth noting that the 1 1B1 state is qualitatively well described even with the smallest generally contracted ANO-L basis sets but not with the standard 6-31G(d,p) basis set, with the 1 1B1 state placed vertically at 7.79 and 8.68 eV, respectively. The latter is more than 1 eV higher than the experimental datum, 7.45 eV.4 Taking into account the calibration calculations collected in Table I, the O[5s4p2d1f] / H[3s2p1d] + (2s2p2d) basis set was used for the calculation of vertical excitation energies, while the smaller O[5s4p1d] / H[2s1p] + (2s2p1d) basis set was employed in the determination of the stretching potential energy curves of the low-lying excited states (basis set: VII and V, respectively, according to the notation used in Table I).

Multiconfigurational wave functions were initially determined at the CASSCF level.41 In all calculations, the oxygen 1s electrons were inactive and all the valence orbitals were included in the active space. The full valence active space

TABLE I. CASPT2 vertical excitation energies (eV) and CASSCF second moments (a.u.) computed for the 1 B, state with different basis sets. Calculations at the experimental ground-state geometry. The full valence active space is used in the calculations.

Basis seta CASPT2 <x2> </> fe2> <r2>

I: 6-31G(d,p) 8.68 5.4 6.6 6.8 18.8

II: [3s2p1d]/[2s1p] 7.79 9.1 9.7 10.3 29.1

III: [4s3p1d] / [2s1p] 7.66 9.2 10.1 10.8 30.1

IV: [5s4p1d] / [2s1p] 7.49 9.2 9.7 10.7 29.6

V: [5s4p1d]/ [2s1p]+s, s' + p,p' + d 7.43 9.7 10.4 12.6 32.7

VI: [5s4p2d1 f] / [3s2p1d] 7.48 9.8 10.3 11.6 31.7

VII: [5s4p2d1 f] / [3s2p1d] + s, s' + p,p' + d, d' 7.48 9.4 10.2 12.3 31.9

VII: [6s5p3d2f] / [3s2p1d] 7.51 9.7 10.2 11.7 31.6

Expt.b 7.447 eV

"Generally contracted basis sets of the ANO type which are obtained from O(14s9p4d3f) / H(8s4p3d) primitive sets. When indicated, the basis set is supplemented with two s (exponents 0.023 48 and 0.008 22), two p (exponents 0.015 79 and 0.005 53), and one or two d (exponents 0.056 27 and 0.019 70) diffuse functions placed at the oxygen atom position, which are constructed from an even-tempered expansion from the primitive set.

bHigh resolution absorption spectrum from Ref. 4. Band maximum.

comprises six orbitals with eight electrons and corresponds to (3120) where the entries stand for the number of active orbitals belonging to the irreducible representations a1, b1, b2, and a2, respectively, of the C2v point group. Rydberg orbitals have to be added to the valence active space in order to have enough flexibility for computing all types (Rydberg, valence, or mixed valence-Rydberg) of excited states. With the simultaneous inclusion of the 3s-, 3p-, and 3d-Rydberg orbitals (nine orbitals), the active space becomes unnecessarily large. Therefore, selected Rydberg orbitals were added to the full valence active space according to the symmetry of the state and taking into account that we are interested in excitations from the two highest occupied molecular orbitals (1b1 and 3a1). For all the states, the computed vertical excitation energies were checked with respect to further enlargements of the active space and stable results (±0.1 eV) were found. The molecular orbitals for the excited states were obtained from state-average (SA) CASSCF computations. Table II compiles the final active spaces employed for the calculation of the singlet and triplet excited states of water and the number of states included in the SA CASSCF calcu-

TABLE II. Details of the CASSCF wave functions used for the calculation of the considered excited states of water.

State Active spacea N b 1' states

1,3A1(3a1 — 4a1,3s ;1b1 — 3px) (4220) 3

1,3A1(1b1 — 3dxZ) (3320) 4 (singlet),

1,3A1(3a1 — 3pz) (5120) 4

1,3Bl(1b1 — 4a1,3s ,3pz) (5120) 3

1,3B1(1b1 - 3d2,3d2, y2) (7120) 5

1,3B1(3a1 — 3pX) (3220) 2

1,3B2(1b! — 3dXy ;3a1 — 2b2,3py) (3141) 2

1,3A2(1b! — 2b2,3py ,3dyZ) (3141) 3

aNumber of active orbitals of the irreducible representations of the C2v point group (a1,b1,b2,a2).

bNumber of roots included in the corresponding state-average CASSCF calculation.

lations. The transition energies are referred to the ground state computed with the same active space as the excited state.

To take into account the remaining (dynamic) correlation effects, the CASSCF wave functions were employed as reference functions in a single-state second-order perturbation CASPT2 treatment.35,36 The oxygen 1s electrons were kept frozen at this calculation level. In addition, the so-called imaginary level-shift technique44 was used with a shift value of 0.2 a.u. in order to avoid the possible presence of weakly interacting intruder states. The coupling of the CASSCF wave functions via dynamic correlation was dealt by using the extended multistate CASPT2 approach, the MS-CASPT2 method.45 An effective Hamiltonian matrix is constructed where the diagonal elements correspond to the CASPT2 energies and the off-diagonal elements introduce the coupling to second order in the dynamic correlation energy. In this way, all the states of a given symmetry are allowed to interact under the influence of dynamic correlation, and the possible erratic valence-Rydberg mixing can be removed (see, for instance, Ref. 46). The linear combination of the CASSCF states produced by the MS-CASPT2 calculation is named perturbatively modified CAS (PMCAS) CI wave function.45 Properties of the states were determined from the PMCAS-CI wave functions. The oscillator strengths were computed using PMCAS-CI transition dipole moments according to the CASSCF state interaction (CASSI) method47,48 and energy differences obtained at the MS-CASPT2 level. All calculations were performed with the MOLCAS-6 quantum-chemistry software.49

III. RESULTS AND DISCUSSION

In C2v symmetry the ground state of the water molecule is mainly described by the electronic configuration (1a1)2(2a1)2(1b2)2(3a1)2(1b1)2 where the 1a1 orbital corresponds to the 1 s orbital of the oxygen atom. The valence orbitals comprise the four highest occupied orbitals listed above and two unoccupied antibonding orbitals of a1 and b2

TABLE III. Computed MS-CASPT2 vertical transition energies (AE, eV), oscillator strengths (f), and second moments (from PMCAS-CI wave functions) (a.u.) for the singlet excited states of the water molecule at the ground-state equilibrium geometry.

State AE f <x2> <y2> <z2> <r2> Expt. (eV)

1A, 5.6 3.3 4.5 13.4

1 1B1(1b1 ^ 3s/4a1) 7.50 0.033 8.9 9.8 11.4 30.1 7.4,a 7.447,b 7.42c

1 1A2(1b1 ^ 3py / 2b2) 9.27 Forbidden 9.8 22.2 11.2 43.2 9.1a

2 1A1(3a1 ^ 3s/4a1) 9.86 0.032 20.1 12.2 12.7 45.0 9.7,a 9.991b

21B1(1b1 ^ 3pz) 9.95 0.006 16.4 16.2 31.3 63.9 10.01,a'b 9.99c

3 1A1(1b1 ^ 3px) 10.15 0.032 33.2 13.7 15.1 62.0 10.16,ab 10.17c

21A2(1b1 ^ 3dyz) 10.84 Forbidden 21.5 55.5 29.0 106.0 10.84a

1B1(1b1 ^ 3d) 11.03 0.010 22.4 48.7 28.6 99.7 10.99d

1 1B2(1b1 ^ 3dxy) 11.05 0.012 37.6 35.8 14.4 87.8 11.05d

1A1(1b1 ^ 3dxz) 11.09 0.024 60.5 20.7 41.4 122.6 11.13,d 11.12c

1B1(1b1 ^ 3d) 11.19 0.015 33.4 21.0 60.8 115.2

2 1B2(3a1 ^ 3py / 2b2) 11.69 0.020 13.1 29.3 12.4 54.8 11.5e

1B1(3a1 ^ 3px) 12.13 0.002 43.2 14.6 15.6 73.4

1A1(3a1 ^ 3pz) 12.21 <0.001 16.5 15.3 35.7 67.5

aElectron-impact data from Chutjian et al. Ref. 16.

bHigh resolution vacuum ultraviolet spectrum from Mota et al., Ref. 4.

cWang et al., Ref. 11.

dIshiguro et al., Ref. 13.

eDutuit et al., Ref. 23.

symmetries (4a1 and 2b2 orbitals), as can be derived from a simple qualitative analysis based on symmetry considerations.27 The highest occupied molecular orbital, 1b 1, is a nonbonding orbital closely related to the 2px orbital of the oxygen atom. This assignment is consistent with the vibrational structure of the lowest-energy band of the photo-electron spectrum.50 The 0-0 feature is the most intense transition of the band, indicating a similar geometry for the neutral molecule and the cation, as could be expected from ionization involving a nonbonding electron. Accordingly, the 1 2B1 state has been computed at the CASPT2 level (valence active space) to lie vertically at 12.54 eV above the ground state, in agreement with both the 0-0 band position recorded at 12.61 eV (Ref. 50) and the value determined from the convergence of the Rydberg series.9,10 The second-energy state of the cation, 1, arising from removing one electron from the 3a1 orbital, has been computed vertically at 14.74 eV. Experimental data show a large energy difference (~0.8 eV) between the vertical and adiabatic ionization energies, suggesting therefore an important geometrical molecular reorganization upon ionization, an effect which is consistent with a linear or quasilinear geometry for the 1 2A1 state.50 As determined from the band maximum, the second vertical ionization energy is 14.7 eV,50 matching the present computed value. According to the experimental ionization potentials,50 the 1b2 orbital is energetically well separated (>3 eV) from the two highest occupied MOs. Therefore, we shall focus the study on the excited states of water arising from excitations from the 3a1 and 1b1 MOs. The 3s-, 3p-, and 3d-members of the singlet and triplet Rydberg series converging to the lowest ionization potential and the 3s- and 3p-Rydberg members converging to the 1 2A1 state have been considered. Different types of equilibrium geometries (bent or linear) are expected for the Rydberg states originating from the 1 b1 and 3a1 orbitals, respectively, which can

have important consequences in the deactivation pathways of the states.51,52 Apart from Rydberg states, from a simple qualitative analysis based on the MOs of the ground state, one could also expect four valence (singlet and triplet) excited states, each one belonging to one of the irreducible representations of the C2v symmetry group, arising from one-electron excitations from the two highest occupied MOs to the valence unoccupied 4a1 and 2b2 orbitals. These possible valence excited states can interact with the corresponding Rydberg states of the same symmetry and close in energy. For instance, the valence singly excited 1b1 ^ 4a1 configuration could be mixed with the 1b1 ^ 3s, 1b1 ^ 3pz, 1b1 ^ 3dx2-y2, and 3dz2 Rydberg states, especially with those energetically closer. As outlined in the Introduction, it seems that the valence-Rydberg mixing occurs at the molecular orbital level in the water molecule and results in an electronic spectrum with predominant Rydberg character at the equilibrium ground-state geometry, as confirmed by the present findings (see Table III).

The computed results and selected experimental and previous theoretical information on the singlet excited states of water at the FC geometry are collected in Tables III and IV. The nature of the states is determined from the corresponding PMCAS-CI wave functions and it is shortly described within parentheses in the tables. Thus, the states have been characterized on the basis of the dominant orbital configuration and the expectation values (x2), (y2), and (z2), which provide an estimate of the diffuseness of the state and the directional distribution of the electronic charge. It is noted that all the states are appreciably more diffuse than the ground state for which the value (r2) is computed to be 13.4 a.u. A detailed analysis of the nature of the states is carried out below. It is also worth mentioning that, except for the 2 1A1 and 3 1A1 states, the computed PMCAS-CI wave functions are similar to those obtained at the CASSCF level.

TABLE IV. Computed MS-CASPT2 (MS-PT2) vertical excitation energies (eV) for the singlet excited states of the water molecule compared to previous theoretical results.

State MS-PT2 MRD-Cla TD-DFTb EOM-CCSDb CCSDR(3)c MRCCd MRCI+Q'

1 1B1(1b1 ^ 3s/4a1) 7.50 7.63 7.61 7.58 7.66 7.51 7.81

1 1A2(1b1 ^ 3py / 2b2) 9.27 9.60 9.36 9.33 9.42 9.30 9.30

2 1A1(3a1 ^ 3s/4a1) 9.86 9.95 9.76 9.86 9.93 9.84 9.91

21B1(1b1 ^ 3pz) 9.95 10.11 10.00 9.99 10.09 10.06 10.72

3 1A1(1b1 ^ 3pJ 10.15 10.11 10.23 10.21 10.28 10.19 11.26

21A2(1b1 ^ 3dyz) 10.84 11.02 10.98 11.06

1B1(1b1 ^ 3d) 11.03

1 1B2(1b1 ^ 3dxy) 11.05 11.11 11.25

1A1(1b1 ^ 3dxz) 11.09 11.16 11.49

1B1(1b1 ^ 3d) 11.19

2 1B2(3a1 ^ 3py / 2b2) 11.69 11.54 11.69 11.76 11.74

1B1(3a1 ^ 3pJ 12.13

1A1(3a1 ^ 3pz) 12.21

aHarrevelt and van Hermert, Ref. 33. bd-aug-cc-pVTZ basis set. Cai et al., Ref. 32. cd-aug-cc-pVQZ basis set, Christiansen et al., Ref. 31. dLi and Paldus, Ref. 6. eaug-cc-pVTZ basis set. Cai et al., Ref. 32.

Accordingly, the vertical excitation energies computed at the MS-CASPT2 level practically coincide (within ±0.03 eV) with those obtained at the CASPT2 level. For the 2 1A1 and 3 A states, the coupling via dynamic correlation has been found to be important since both states are found within an energy difference of 0.13 eV at the CASPT2 level. The MS-CASPT2 calculation changes the relative order of these states with respect to that obtained at the CASPT2 level. Moreover, the PMCAS-CI wave functions of the 2 1A1 and 3 1A1 states show a higher degree of configurational mixing involving the singly excited 3a1 ^ 3s/4a1 and 1b1 ^ 3px configurations than that obtained for the CASSCF wave functions, in agreement with recent CAS/multireference configuration interaction (MRCI) calculations reported by Borges. At first glance, the results collected in Table III deviate from the experimental values less than 0.1 eV, excluding the computed vertical excitation energies for the three lowest excited states and the 2 1B2 state which show a somewhat larger difference (0.2 eV). In the following, we shall first discuss in detail the results obtained for the singlet excited states in comparison with the experimental data and previous ab initio results. Next, the triplet excited states will be considered.

Singlet excited states. As shown in

Table III, the 1 1B1

state is computed to be the lowest singlet excited state of the water molecule, in agreement with previous theoretical results (see Table iv).6,8,31,32 The 1 1B1

state is vertically

placed at 7.50 eV and it is related to one of the most intense transitions of the electronic spectrum. The computed oscillator strength of 0.033 has the same order of magnitude as the

value (0.049) obtained by Borges7 at the CAS/MRCI level

and the experimental determinations. It is well established that the 1 1B1 state, usually called A state, is responsible for the lowest-energy band of the electronic spectrum with maximum at around 7.4 eV. , , , , The nature

of the 1 1B1

state needs, however, further consideration. The band is

broad, covering the energy range 6.8-8.2 eV, and has a poorly defined progression which has been related to the symmetric bending mode4,11 or to the symmetric stretching vibration on the basis of the results reported by Makochen-kawa et al.54 Comparison with the lowest-energy band of the photoelectron spectrum,50 corresponding to the removal of a 1b1 electron, showed that both bands have different shapes, suggesting therefore some valence character for the 1 1B1 state, according to Robin.14 On the contrary, a more detailed analysis of the band at 7.4 eV performed by Wang et al.11 led these authors to propose a dominant valence nature for the 1 1B1 state. Furthermore, the corresponding 1B1 (1b1 ^ 3s) state with predominant Rydberg character was tentatively located at 9.08 eV.11 The present results, as well as previous theoretical studies (see Table IV), clearly rule out the existence of a 1B1 state near 9 eV and suggest a valence-Rydberg mixing for the 1 1B1 state. On one hand, considering the computed (r2) values included in Table III, we find that the

1 1B1 state has a diffuse nature ((r2) = 30.1 a.u.), more than twice the value calculated for the ground state but appreciably lower than that obtained for a pure Rydberg state as the

2 1B1(1b1 ^3pz) state ((r2) = 63.9 a.u.). As shown in Table I, it is worth recalling that the computed (r2) is quite stable with respect to the inclusion of Rydberg functions in the basis set. On the other hand, the PMCAS-CI wave function computed for the 1 1B1 state is mainly described by the singly excited 1b1 ^ 3s/4a1 configuration with a weight of 75%. Therefore, the valence-Rydberg character of the state arises mainly from the nature of the excited orbital, labeled here as 3s/4a1 to illustrate its intermediate character, and not from configurational mixing as seems to suggest the results reported by Diercksen et al.22 Thus, we see that the excited orbital is not a pure 3s-Rydberg orbital but it has a valence contribution, as reflected by the calculated value for (r2) and

it was also obtained in previous theoretical studies.34,55 At the ground-state equilibrium geometry, the present MS-

TABLE V. Computed MS-CASPT2 excitation energies (eV) and other properties for the singlet excited states of symmetrically stretched (r0-H= 1.4658 A) water. Results from calculations with the basis set V and active space (5120) for the 1B1 states, (4220) for the 1A1 states, and (3141) for the 1A2 states.

State AEa (x2) (y2) (z2) (r2)

1 1B1(1b1 ^ 4a1) 7.90 5.8 5.8 6.6 18.2

1 1A2(1b1 ^ 2b2) 9.11 5.1 5.9 5.8 16.8

2 1A1(3a1 ^ 4a1) 11.21 7.5 5.3 5.2 18.0

2 1B1(1b1 ^ 3 s) 13.00 27.3 18.8 31.4 77.5

3 1B1(1b1 ^ 3pz) 13.19 17.6 14.5 32.2 64.3

3 1A1(1b1 ^ 3px) 13.22 33.1 9.8 12.6 55.5

21A2(1b1 ^ 3py) 13.43 21.5 51.0 19.7 92.2

aExcitation energies are referred to the ground state computed at the experimental equilibrium geometry (rO-H=0.9578 A).

CASPT2 calculations predict only one 1 ^3s/4a^

state of intermediate type. These results strongly support the consideration of the 1 1B1 state of water as a case of MO Rydbergization postulated by Mulliken.21'28

In order to get further insight into the nature of the 1 1B1 state, we have analyzed its evolution with respect to the O-H internuclear distance. For the sake of simplicity, we have considered the symmetrical stretching of both O-H bonds, preserving thus the C2v symmetry of the system. MS-CASPT2 calculations were performed for the three lowest roots of 1B1 symmetry using the O\5s4p1d] / H\2s1p] + (2s2p1d) basis set (basis set V) and the (5120) active space (full valence active space plus two extra Rydberg orbitals of a1 symmetry) at the C2v geometries with O-H distances varying between 0.916 and 1.466 Á and keeping fixed the bond angle to the ground-state equilibrium value (104.5°). The excitation energies and second moments computed for the singlet excited states of water at the O-H internuclear distance of 1.466 Á are included in Table V. The computed potential energy curves are shown in Fig. 1 together with the values for (r2), which allow us to characterize the states. The computed vertical excitation energies at the ground-state equilibrium geometry (rO-H = 0.958 Á) are only 0.04-0.1 eV lower than those obtained with the larger basis set VII (see Table III), supporting then the basis set used for building the potential energy curves. As expected from the results collected in Table III, the three low-lying 1B1 states are related to one of the 1b1 ^ 3s/4a1, 1b1 ^ 3pz, and 1b1 ^ 4s one-electron promotions at geometries with O-H bond lengths around the ground-state equilibrium value (r ~ 0.9-1.0 Á)

(see also (r2) values). It should be clarified that the 1B1(1b 1 ^ 4s) state is not included in Table III although it was computed to appear in the energy range studied (vertically located at 10.55 eV with the basis set VII). As stated in the Introduction, the present study is focused on the lowest members (n=3) of the Rydberg series and more diffuse functions should be probably included in the basis set to get accurate results for the n=4 members. Our calculations are expected, however, to yield a reliable description for the 1B1(1b 1 ^ 4s) state, which is here considered in order to get a more complete picture of the evolution of the 1B1 states as function of the O-H internuclear distance. As shown in Fig. 1, the second and third 1B1 states have Rydberg character at all distances examined since they show high values for (r2) in comparison with the ground state. The type of Rydberg state changes, however, from 1b1 ^ 3pz and 1b1 ^ 4s to 1b1 ^ 3 s and 1b1 ^ 3pz, respectively, when increasing the O-H distances. The shape of the corresponding potential energy curves is similar to the ground-state curve, as could be expected for Rydberg states converging to the 2B1 cationic state which has a geometry close to the ground-state one.50 As can be seen in Fig. 1, the situation is clearly different for the 1 1B1 state. The evolution of the computed (r2) for the 1 1B1 state with the O-H distance shows that the state is getting more valence character when stretching the O-H bond as a result mainly of the changes undergone by the virtual orbital, in agreement with the concept of Rydbergization.21,28 From a simple qualitative description, the energy of the antibonding 4a1 orbital decreases when

FIG. 1. (Color online) Symmetrical stretching of the O-H bonds (<HOH 104.5°) in the ground and low-lying 1 B1 states of the water molecule. Potential energy curves referred to the corresponding ground-state minimum (left) and computed (r2) values (right).

FIG. 2. (Color online) Symmetrical stretching of the O-H bonds (<HOH 104.5°) in the ground and low-lying 1A2 states of the water molecule. Potential energy curves referred to the corresponding ground-state minimum (left) and computed (r2) values (right).

lengthening the O-H distance separating this orbital from the related 3s-Rydberg orbital. Accordingly, the second 1B1 root corresponds to the 1b1 ^ 3s-Rydberg state at O-H distances longer than 1.32 A (see Table V). The nature of the 1 1B1 state evolves then from a valence-Rydberg mixing type (1 b 1 ^ 3s/4a1) at the ground-state equilibrium geometry to a valence character (1b1 ^ 4a1) at O-H bond lengths around 1.25 A and longer, which corresponds only to a symmetrical bond stretching of 0.3 A from the ground-state equilibrium value. The same evolution is expected when stretching only one O-H bond length, with the valence character of the state coming from the O-H a* orbital, as it has been corroborated by actual calculations. Therefore, the present work confirms definitely that the 1 1B1 state of the water molecule constitutes an example of MO Rydbergization postulated by 21 28

Mulliken. The reported change in the nature of the state does not arise from a crossing between separated valence and Rydberg states. It exists only one state with changing nature.

The second singlet 1 1A2 excited state is related to a dipole-forbidden transition. Experimental evidence for this state comes from electron-impact spectroscopy and gives an excitation energy of 9.1 eV.16 The experimental datum is somewhat lower than the MS-CASPT2 vertical excitation energy, computed to be 9.27 eV, and than previous theoretical estimations 6,31-33 (see Table IV). According to the computed (r2) value, the 1 1A2 state is predicted to be a Rydberg state but with some valence character which occurs at the molecular orbital level, similarly to the 1 1B1 state. In this case, the 3py-Rydberg orbital (b2 symmetry) has certain contribution from the valence antibonding 2b2 orbital and the singly excited 1b1 ^ 3py / 2b2 configuration appears in the computed PMCAS-CI wave function for the 1 1A2 state with a weight of 73%. Two characteristics show the valence-Rydberg nature of the 1 1A2 state: first, its computed (r2) has a lower value than the one obtained for the pure 1b1 ^ 3pz Rydberg state (43.2 vs 63.9 a.u.) and second, the 1 1A2 state is separated by ~0.7 eV from the remaining 1b1 ^ 3p Ryd-berg states, an aspect attributed by Robin14 to the stabilization of the 3py orbital through the mixing with the valence 2b2 orbital. Despite both results were also found in the ear-

lier study carried out by Goddard and Hunt, these authors attributed a Rydberg character to the 1 1A2 state. Further insight has been obtained here by means of the analysis of the effect of the O-H stretching on the energy and (r2) of the

low-lying 1A2 states, similarly as it was performed for the 1 1B1 state. We have also considered symmetrical O-H stretching, keeping fixed the bond angle to the ground-state equilibrium value and preserving thus the C2v symmetry of the system. Calculations were performed using the basis set V and the (3141) active space with eight active electrons. The results are shown in Fig. 2 and Table V for the longest O-H distance considered. As can be deduced from the (r2) values included in Fig. 2, the diffuse nature of the 1 1A2 state decreases when increasing the O-H distance, while the second computed 1 A2 state is a Rydberg state at all the distances considered but varies the size of the Rydberg orbital. The evolution of the 1 1 A2 state with the O-H distance, as well as the origin of the transformation found, are the same as those described for the 1 1B1 state, but affecting now the valence antibonding 2b2 orbital and the 3py-Rydberg orbital. The nature of the 1 1A2 state evolves then from a valence-Rydberg mixing (1b1 ^ 3py / 2b2) at geometries close to the ground-state equilibrium geometry to a valence character (1 b1 ^ 2b2) at O-H distances longer than —1.217 A where the second 1A2 state becomes the 1b1 ^ 3py Rydberg state (see Table V). As shown in Fig. 2, the potential energy curve computed for the 1 1 A2 state is very flat and shows a minimum at O-H distances around 1.11 -1.16 A. Symmetrically stretching the O-H bond length by —30% from its equilibrium value transforms the 1 1A2 state in a pure valence excited state. This state represents therefore another example of

Rydbergization at the MO level, and such change in nature does not come from any crossing between the computed 1 A2 potential energy curves, as shown in Fig. 2.

The second band of the electronic spectrum of water in the gas phase appears as a broad and continuous absorption covering the energy range 8.6-10.0 eV and peaking at

9.7 eV.4,11,12 The band has superposed a long vibrational pro-

gression attributed firstly to the bending mode and more recently to both bending and symmetrical stretching vibra-tional modes.4 On the basis of the vibrational analysis of the band and the earlier ab initio calculations,17,18 it was assigned to the so-called B state, corresponding to the 2 1A1 state mainly characterized by the 3a1 ^ 3s excitation. The long vibrational progression on the bending mode indicates an important change in the bond angle upon excitation, an effect which is consistent with electron promotion from the 3a1 orbital.11 Calculations on the bending potential energy

FIG. 3. (Color online) Symmetrical stretching of the O-H bonds (<HOH 104.5°) in the ground and low-lying 1A1 states of the water molecule. Potential energy curves referred to the corresponding ground-state minimum (left) and computed (r2) values (right).

1 29,33,52

curves predicted a linear geometry for the 2 1A1 state. ' '

On the other hand, excitation to the second band has been

established to lead also to photodissociation through a more

complex mechanism33,51,52 than in the lowest absorption

band. As shown in Table III, the MS-CASPT2 results predict the 2 1A1 state to be situated vertically at 9.86 eV above the ground state and support its assignment to the second band of the spectrum. Mota et al.4 have recently reported an experimental value of 9.991 eV for the origin of the Rydberg series arising from excitation from the 3a1 orbital. Although it is unclear for us from their discussion,4 it seems that they attribute the second band of the spectrum to a 1A1 state with valence character and place the 1A1 (3a1 ^ 3s) Rydberg state on the high-energy side of the band. Such interpretation would be wrong according to the present results and previous theoretical studies31-33 (see Table IV) since theory predicts only one 1A1 state in the energy range of the second absorption band. Although we have characterized the 2 1A1 state mainly by means of the 3a1 ^ 3s/4a1 promotion, this state is described by a mixing of the singly excited 3a1 ^ 3s/4a1 and 1b 1 ^ 3px configurations, with weights of 70% and 28%, respectively. The computed (x2), (y2), and (z2) values for the 2 1 A 1 state reflect the configurational mixing (see Table III). As discussed above, the 2 1A1 state is then closely related to the 3 1A1 state which is characterized by the same configurations but in a different proportion (weights of 69% and 28% for the 1b1 ^ 3px and 3a1 ^ 3s/4a1 one-electron promotions, respectively). A similar description for the 2 1A1 and 3 1A1 states was also reported in ^previous theoretical studies with MRCI methodologies. , , It should be noted here that the one-electron 1b1 ^ 3px promotion corresponds to a pure Rydberg excitation since the molecule lacks virtual valence orbitals of b1 symmetry. In contrast, as indicated by the notation used, the 3a1 ^ 3s/4a1 excitation has a valence-Rydberg mixed character because of the nature of the excited orbital. The configurational mixing increases therefore the Rydberg character of the 2 1A1 state at the ground-state equilibrium geometry, as reflected by the computed (r2). However, some valence contribution in the B 2 1A1 state, ascribed to the 3s/4a1 orbital, is still present and it has been suggested from the experimental studies about photodissociation processes of water.23 As it was found for the 1 1B1 state (see Fig. 1), the excited orbital (3s/4a1) is expected to increase its valence character when elongating the O-H bond. This evolution of the 3s/4a1 orbital was obtained in an earlier

study of the 2 1A1 state performed at the SCF level by Hors-

ley and Fink. The authors classified then the state as a

semi-Rydberg state following Mulliken's description. To

our knowledge, no further analysis of the nature

states have been reported, although their potential energy

curves have been computed.33,52,56 To shed more light into

the nature of the low-lying 1A1 states, we have computed the three lowest 1A1 states at C2v geometries with O-H distances going from 0.916 to 1.466 A and focused our attention on the variations of the computed (r2) values with the O-H distance. Calculations were carried out using the basis set V and the active space (4220), namely, the minimal active space required to describe the states of interest, and the bond angle was fixed to the ground-state equilibrium value. The results are collected in Fig. 3 and Table V. The evolution of the computed (r state shows that the valence

character of the state increases when stretching the O-H bonds. It is due to the enlargement of the weight of the singly excited 3a1 ^ 3s/4a1 configuration in the PMCAS-CI wave function of the 2 1A1 state when elongating the O-H bond together with the transformation of the excited orbital toward a valence 4a1 orbital. For instance, the 3a1 ^ 3s/4a1 configuration appears with a weight of 82% at rO-H= 1.064 A (14% for the singly excited 1b1 ^ 3px configuration), that is, near to the minimum of the potential energy curve of the 2 1A1 state (see Fig. 3). As expected, the reduction of the configurational mixing as increasing the O-H distance also affects the 3 1A1 state and it is reflected in the evolution of the computed (r2) values. Thus, the 3 1A1 state becomes progressively a pure 1b1 ^ 3px-Rydberg state when lengthening the O-H bond in the distance range considered. In summary, the present calculations have characterized the 2 1A1 state as a valence-Rydberg state at the ground-state equilibrium geometry and they have shown that the mixing is not only due to the nature of the excited orbital (3s/4a1) but also to configurational mixing involving the Rydberg 1b1 ^ 3px excitation. The situation is then different from that found for the two lowest singlet excited states (1 1B1 and 1 1A2) where the valence-Rydberg character arises from MO Rydbergization. On the other hand, the 3 1A1 state can be considered a Ryd-berg state with a somewhat minor valence character.

Regarding the higher-energy electronic spectrum of gasphase water, it consists of a series of sharp peaks in the 10.0-10.8 eV energy range.4,11,12 The 1b1 ^ 3pz and 1b1

— 3px transitions have been recorded at 10.01 and 10.16 eV in a recently reported high resolution spectrum,4 in agreement with previous experimental determinations.11,12,16 These transitions are related to the so-called C and D

states, which correspond, respectively, to the 2 B1 and 3 1A1 states. The 1b —>4s transition has also been placed in this energy range around 10.6 eV.4,12 As shown in Table III, the vertical excitation energies computed for the 2 1B1 and 3 1A1 states are 9.95 and 10.15 eV, respectively, confirming the assignment of the observed peaks.4,11,12,16 The oscillator strength for the 1 1A1 — 2 1B1 transition is computed to be 0.006, in agreement with recent high-level theoretical determinations.8,31 This low value is consistent with the Rydberg nature of the state and with the selection rule A1 = ± 1 for atomic Rydberg transitions which can be applied here in an approximate way because the 1b 1 orbital is closely related to an oxygen 2p orbital. The rule explains, for instance, the higher intensity computed for the 1b1 — 3d transitions (see Table III). On the other hand, the transition to the D 3 1A1 state is predicted here to have the same intensity as the 1 1A1 — 2 1A1 transition, in contrast to the results obtained at the coupled-cluster and CAS/MRCI level which characterize the 1 1A1 — 2 1A1 transition as the most intense one8,31 but quite similar to the 1 1A1 — 3 1A1 transition (values of 0.055 and 0.047 at the CAS/1MRCI le1vel8). The computed oscillator strengths are, in general, of the same order of magnitude than the experimental values (see Refs. 7 and 8 for a detailed analysis).

The 1b1 — 3d Rydberg transitions have been experimentally located in the 10.8-11.2 eV energy range,11-14,16 which is definitely confirmed by the present MS-CASPT2 results (see Table III). Thus, the computed (r2) values as well as the directionality of the electronic charge distribution characterize the singlet excited states calculated between 10.8 and 11.2 eV as the five components of the 1b1 — 3d Rydberg promotions. The dipole-forbidden transition to the 2 1A2 state, related to the 1b1 — 3dyz one-electron promotion, is computed vertically at 10.84 eV. The 2 1A2 state has been detected by electron-impact spectroscopy which has provided an excitation energy of 10.84 eV,16 matching the present MS-CASPT2 result. Gurtler et al.12 assigned the sharp and structured bands at around 11.0 eV to the four optically allowed 1 b1 — 3d transitions on the basis of their rotational contours. Assignments based on the defect quantum number analysis also situate the 3d-members of the Ry-dberg series converging to the lowest ionization potential at these energies.11,13 In particular, Ishiguro et al.13 recorded three bands with maximum at 10.99, 11.05, and 11.13 eV which were ascribed to the optically allowed 1b1 — 3d Ryd-berg states. According to the present calculations, these three bands correspond, respectively, to the 1B1(1b1 — 3d), 1 1B2(1b1 — 3dxy), and both 1A1(1b1 — 3dxz) and 1B1(1b1

— 3d) states.

The 3a1 — 3p Rydberg transitions are related to states of 1B2(3a1 — 3py), 1B1(3a1 — 3px), and 1A1(3a1 — 3pz) symmetries. As can be seen in Table III, they are computed to be vertically at 11.69, 12.13, and 12.21 eV above the ground state. The 2 1B2 state appears well separated, 0.45 eV lower

in energy, from the other two 3a1 — 3p states. Moreover, the computed (r 2) for the 2 1B2 state is lower than the (r2) values obtained for the 1B1(3a1 — 3px) and 1A1(3a1 — 3pz) states (54.8 vs 73.4 and 67.5 a.u., respectively). These two features point out to the mixed valence-Rydberg nature of the 2 1B2 state. The mixing of the 3py-Rydberg orbital and the valence antibonding 2b2 orbital, allowed by symmetry, is responsible for the valence-Rydberg character, as revealed by the analysis of the PMCAS-CI wave function of the 2 1B2 state. The situation is therefore similar to that found for the 1 1A2(1b1

— 3py/ 2b2) state. The partial valence character can also explain the relative high intensity predicted for the 1 A1

— 2 1B2 transition. Compared to the oscillator strengths computed for the 3a1 — 3px and 3a1 — 3pz Rydberg transitions, that of the 1 1A

>2 B2 transition is more than ten times

larger. Experimental evidence for the 2 1B2 state comes from a study performed by Dutuit et al23 These authors proposed the 2 1B2 state as responsible for the continuous absorption recorded in the energy range 10.0-12.6 eV, centered at around 11.5 eV. Taking into account the great complexity of the electronic spectrum of water at energies above

11.0 eV, together with the low intensity values predicted for the 3a1 ^ 3px and 3a1 ^ 3pz Rydberg transitions, their experimental locations have not been determined, as far as we know. According to the MS-CASPT2 results, both states are located close to the lowest ionization potential of the molecule (—12.6 eV),50 where the absorption spectrum is dominated by the 1b1 ^ nd transitions.20

We will finally discuss briefly some aspects about previous theoretical studies on the excited states of water. The most recent and representative results are included in Table IV. When comparing these previous results to the present MS-CASPT2 data, it should be kept in mind that not all of them have used the experimental ground-state equilibrium geometry for the calculations as in our study. An exhaustive recompilation of theoretical data for the excited states of water can be found in a recent study reported by Li and Paldus.6 These authors chose the water molecule to analyze the performance of their multireference coupled-cluster method (MRCC results in Table IV). Similarly, Cai et al.32 have studied the excited states of water using time-dependent density functional theory (TD-DFT) in order to determine what characteristics of the potential are required to get results of quality comparable to those obtained with the equation of motion coupled cluster (EOM-CC) and MRCI techniques. The TD-DFT values included in Table IV correspond to the best TD-DFT results reported by Cai et al.32 and they have been obtained using an asymptotic correction in the density functional, which has been shown to be necessary in order to get reliable results for states with a significant Rydberg character.32 For the five lowest singlet excited states, the CCSDR(3) vertical excitation energies (noniterative triples correction to CCSD excitation energies) reported by Chris-

tiansen et al., which probably represent the highest level of theory used, are 0.07-0.16 eV larger than the present MS-CASPT2 results. In contrast, the CCSDR(3) vertical excitation energies for the 1b1 ^ 3d Rydberg states are 0.2-0.3 eV overestimated with respect to our MS-CASPT2 results and

the experimental data. A better agreement is, however,

TABLE VI. Computed MS-CASPT2 (MS-PT2) vertical excitation energies (eV) and other properties for the triplet excited states of the water molecule compared to experimental data and previous theoretical results.

State MS-PT2 <x2> <y2> <z2> <r2> Expt. (eV)a Theor.

1 3B1(1¿1 ^ 3s/4aj) 7.13 8.2 9.0 10.3 27.5 7.0 7.23,b 7.18,c 7.14d

1 3A2(1¿1 ^ 3py / 2b2) 9.12 9.1 20.1 10.5 39.7 8.9 9.15,b 9.17,c 9.42d

1 3A1(3a1 ^ 3s/4a1) 9.47 9.0 9.0 9.9 27.9 9.3 9.30,b 9.47,c 9.78d

23A1(1b1 ^ 3px) 9.87 35.5 12.4 14.3 62.2 9.81 9.76,b 9.76,c 10.52d

23B1(1b1 ^ 3pz) 9.90 15.9 15.4 30.3 61.6 9.98 9.89,b 9.93,c 10.61d

23A2(1b1 ^ 3dyz) 10.80 21.4 55.1 27.1 103.6 10.68

3B1(1b1 ^ 3d) 11.01 22.8 48.6 24.2 95.6

1 ^B2(1bt ^ 3d,,) 11.04 37.2 35.3 14.3 86.8

3A1(1b1 ^ 3dJ 11.10 68.1 23.2 41.8 133.1

3B1(1b1 ^ 3d) 11.16 32.0 21.3 67.1 120.4

2 3B2(3a1 ^ 3py/ 2b2) 11.41 11.2 24.0 11.0 46.2 11.29,b 11.35,c 11.41d

3A1(3a1 ^ 3pz) 11.96 15.8 14.4 27.6 57.8

3Bpa1 ^ 3px) 12.08 41.6 14.1 15.1 70.8

aElectron-impact data from Chutjian et al., Ref. 16. bTD-DFT/d-aug-cc-pVTZ results from Cai et al., Ref. 32. cEOM-CCSD/d-aug-cc-pVTZ results from Cai et al., Ref. 32. dMRCI+Q/aug-cc-pVTZ results from Cai et al., Ref. 32.

found between the EOM-CCSD and MS-CASPT2 results (see Table IV). Regarding the MRCI results, apart from those included in Table IV, it should be mentioned here the studies on the oscillator strengths of the low-lying electronic transi-

tions of water performed by Borges using the CAS/MRCI

approach. At the highest level of theory used by Borges, the computed vertical excitation energies for the A, B, and D states (7.69, 9.92, and 10.28 eV) differ by less than 0.2 eV from the MS-CASPT2 values (see data for the 1 1B1, 2 1A1, and 3 1A1 states in Table IV). His results for the remaining states examined show, however, a large difference (0.3-0.7 eV). As a whole, the theoretical values included in Table IV for the five lowest excited states are at most in a

0.3 eV energy interval, except for the MRCI+Q results which are clearly overestimated for the highest states.

Triplet excited states. The results obtained for the triplet excited states of water at the FC geometry are shown in Table VI. As can be deduced from the results collected, some of the features found for the singlet excited states are also present in the corresponding triplet excited states. In particular, the computed (r2) values for the three lowest triplet states, 1 3B1, 1 3A2, and 1 3A1, indicate that these states undergo also MO Rydbergization, as it was determined for the corresponding singlet states (see Table III). It is also noted that the 1 3A2(1fc1 ^ 3py/2b2) state appears at 0.7 eV lower in energy than the other two triplet 1b1 ^ 3p Rydberg states, which can be considered as indicative of its valence contribution, as it was discussed for the 1 1A2 state.14 As shown in Table VI, such energy difference also occurs for the triplet 3a1 ^ 3p states. The 2 3B2 state, described by the 3a1 ^ 3py / 2b2 one-electron promotion, is computed to be situated ~0.6 eV below the next 3A1(3a1 ^ 3pz) state. The computed (r 2) for the 2 3B2 state is appreciably lower than those calculated for the 3B1(3a1 ^ 3px) and 3A1(3a1 ^ 3pz) states (46.2 vs 70.8 and 57.8 a.u., respectively). This result suggests a mixed valence-Rydberg nature of the 2 3B2 state, which also arises from MO Rydbergization according to the computed PMCAS-CI wave function.

As could be expected, the computed singlet-triplet energy splittings are, in general, found to be larger (0.3 -0.4 eV) for those states having a mixed valence-Rydberg nature, except for the 1 3A2 state calculated at 9.12 eV, which is 0.15 eV below the corresponding 1 1A2 state (see Table III). The 2 3A1(1b1 ^ 3px) Rydberg state also has a significant singlet-triplet energy splitting (0.28 eV). The 1b1 ^3d Rydberg states show negligible (~0.01 -0.04 eV) singlet-triplet energy splittings. On the other hand, unlike the singlet 2 1A1 and 3 1A1 states, no con-figurational mixing has been found for the related 1 3A1 and 2 3A1 states, as can be inferred from the computed (x2), (y2), and (z2) values for the 1 3A1 state (see Table VI).

As regards the comparison with the experimental data, it is noted that the MS-CASPT2 vertical excitation energies computed for the three lowest triplet excited states are approximately 0.2 eV overestimated with respect to the electron-impact data reported by Chutjian et al.16 The situation is then similar to that found for the singlet excited states with valence-Rydberg character. As can be seen in Table VI, the difference between the MS-CASPT2 results and the experimental data is, however, minor for the three next triplet excited states (2 3B1, 2 3A1, and 2 3A2) having all them Ryd-berg character. Table VI also includes the theoretical values

obtained by Cai et al. for the triplet states using different methods. As a whole, they differ at most 0.3 eV from the MS-CASPT2 vertical excitation energies, except for the MRCI+Q results for the 2 3B1 and 2 3A1 states which are clearly overestimated, as it was found for the corresponding singlet states (see Table IV).

IV. SUMMARY AND CONCLUSIONS

We have presented here a high-level ab initio study on the excited states of the water molecule using the extended multistate CASPT2 method35,36,45 and highly accurate ANO-type basis sets. The study includes the singlet and triplet excited states arising from excitations from the two highest

occupied molecular orbitals (3a1 and 1b1) to the valence and 3s-, 3p-, and 3d-Rydberg orbitals. The states have been characterized by means of the computed PMCAS-CI wave functions and the second moments of the electronic charge distribution. The computed vertical excitation energies have been discussed in relation to experimental data and previous theoretical results. As a whole, the largest deviation (^0.2 eV) from the experimental data has been obtained for the three low-lying singlet and triplet excited states, which can be related to the shape of the recorded bands as well as to the nature of the states. It is concluded that the electronic spectrum of gas-phase water has a predominant Rydberg character at the Franck-Condon geometry, in line with the general view, although a certain degree of valence character has been shown to occur in the three lowest singlet states. Moreover, analysis of the nature of the low-lying excited states through the calculation of a portion of the symmetrical stretching O-H potential energy curves has clearly confirmed

that the valence-Rydberg mixing of the states results from

MO Rydbergization as described by Mulliken. ' Accordingly, there is only one state whose character changes with the molecular geometry as reflected indeed by all our attempts to find pure valence states at the FC geometry. In particular, for the lowest singlet A 1 1B1 excited state, related to the one-electron 1b1 — 3s/4a1 promotion, it has been established that the excited orbital has a substantial 3s Rydberg character at the FC region but evolves to become a valence antibonding 4a1 orbital at elongated O-H bond lengths. This behavior is consistent with the photodissociation process into ground-state H- and OH- radicals following excitation of wa-

ter in the energy range of the lowest absorption band. A valence-Rydberg mixing is therefore also expected for those states ascribed to promotions from deeper orbitals to the

3s/4a1 orbital, as pointed out by Dutuit et al. and it occurs for the 2 1A1 (3a1 — 3s/4a1) state considered here. Excitation to the 3py /2b2 orbital also leads to excited states of valence-Rydberg nature, as shown for the 1 1A2(1b1 — 3py/2b2) state. Finally, it is also worth mentioning that the electronic spectra of CH4 and NH3 molecules have been also interpreted as arising from Rydberg states at the FC region, having the lowest singlet excited state a valence-Rydberg nature.14,20 No valence excited states have been characterized in these molecules at the ground-state equilibrium geometry. As for the water molecule, an increase of the valence character of the

lowest singlet excited state occurs in these hydrides upon

stretching of one of the bonds. Interestingly, Mulliken considered the lowest singlet excited state of the three molecules as an example of MO Rydbergization, although his prediction failed for other systems such as ethene.14,45,57 The present study has confirmed the MO Rydbergization in the lowest singlet 1 1B1 state of the water molecule. In addition, a similar situation has been found for the 1 1 A2 singlet excited state. A more complex behavior has been determined for the 2 1A1 state, where both MO Rydbergization and con-figurational mixing take place. Therefore, whereas at the ground-state equilibrium distance the three excited states have a predominant diffuse character, they have a clear valence nature upon symmetric stretching of the O-H bond, and it is only then when the typical view of states and orbit-

als displayed in general chemistry textbooks is applicable. From the overall study, the essentials of the lowest excited states of the water molecule are therefore better understood.

ACKNOWLEDGMENTS

Financial support is acknowledged from projects CTQ2004-01739 and CTQ2007-61260 of the Spanish MEC/ FEDER and GV06-192 of the Generalitat Valenciana.

R. S. Mulliken, J. Chem. Phys. 3, 506 (1935). W. C. Price, J. Chem. Phys. 4, 147 (1936).

P. Maksyutenko, J. S. Muenter, N. F. Zobov, S. V. Shirin, O. L. Polyan-sky, T. R. Rizzo, and O. V. Boyarkin, J. Chem. Phys. 126, 241101 (2007).

R. Mota, R. Parafita, A. Giuliani, M.-J. Hubin-Franskin, J. M. C.

Lourengo, G. Garcia, S. V. Hoffmann, N. J. Mason, P. A. Ribeiro, M.

Raposo, and P. Limao-Vieira, Chem. Phys. Lett. 416, 152 (2005).

M. López del Puerto, M. L. Tiago, I. Vasiliev, and J. R. Chelikowsky,

Phys. Rev. A 72, 052504 (2005).

X. Li and J. Paldus, Mol. Phys. 104, 661 (2006).

I. Borges, Jr., J. Phys. B 39, 641 (2006).

I. Borges, Jr., Chem. Phys. 328, 284 (2006).

K. Watanabe and A. S. Jursa, J. Chem. Phys. 41, 1650 (1964).

D. H. Katayama, R. E. Huffman, and C. L. O'Bryan, J. Chem. Phys. 59, 4309 (1973).

H. T. Wang, W. S. Felps, and S. P. McGlynn, J. Chem. Phys. 67, 2614 (1977).

P. Gürtler, V. Saile, and E. E. Koch, Chem. Phys. Lett. 51, 386 (1977).

E. Ishiguro, M. Sasanuma, H. Masuko, Y. Morioka, and M. Nakamura, J. Phys. B 11, 993 (1978).

M. B. Robin, Higher Excited States of Polyatomic Molecules (Academic, New York, 1985), Vol. 3.

S. Trajmar and R. I. Hall, J. Phys. B 7, L458 (1974).

A. Chutjian, R. I. Hall, and S. Trajmar, J. Chem. Phys. 63, 892 (1975). W. A. GoddardIII and W. J. Hunt, Chem. Phys. Lett. 24, 464 (1974). N. W. Winter, W. A. Goddard III, and F. W. Bobrowicz, J. Chem. Phys. 62, 4325 (1975).

R. J. Buenker and S. D. Peyerimhoff, Chem. Phys. Lett. 29, 253 (1974). M. N. R. Ashfold and S. R. Langford, in The Role of Rydberg States in Spectroscopy and Photochemistry, edited by C. Sándorfy (Kluwer, Dordrecht, 1999), Chap. 2, p. 23. R. S. Mulliken, Acc. Chem. Res. 9, 7 (1976).

G. H. F. Diercksen, W. P. Kraemer, T. N. Rescigno, C. F. Bender, B. V. Mckoy, S. R. Langhoff, and P. W. Langhoff, J. Chem. Phys. 76, 1043 (1982).

O. Dutuit, A. Tabche-Fouhaile, I. Nenner, H. Frohlich, and P. M. Guyon, J. Chem. Phys. 83, 584 (1985).

V. Engel, V. Staemmler, R. L. Vander Wal, F. F. Crim, R. J. Sension, B. Hudson, P. Andresen, S. Hennig, K. Weide, and R. Schinke, J. Phys. Chem. 96, 3201 (1992).

X. F. Yang, D. W. Hwang, J. J. Lin, and X. Ying, J. Chem. Phys. 113, 10597 (2000).

R. van Harrevelt and M. C. van Hemert, J. Chem. Phys. 114, 9453 (2001).

G. Herzberg, Molecular Spectra and Molecular Structure III (van Nos-trand, New York, 1966).

R. S. Mulliken, Chem. Phys. Lett. 46, 197 (1977). J. A. Horsley and W. H. Fink, J. Chem. Phys. 50, 750 (1969). M. Chergui and N. Schwentner, Chem. Phys. Lett. 219, 237 (1994). O. Christiansen, T. M. Nymand, and K. V. Mikkelsen, J. Chem. Phys. 113, 8101 (2000).

Z. Cai, D. J. Tozer, and J. R. Reimers, J. Chem. Phys. 113, 7084 (2000). R. van Harrevelt and M. C. van Hemert, J. Chem. Phys. 112, 5777 (2000).

D. M. Chipman, J. Chem. Phys. 122, 044111 (2005). K. Andersson, P.-Ä. Malmqvist, B. O. Roos, A. J. Sadlej, and K. Wolinski, J. Phys. Chem. 94, 5483 (1990).

K. Andersson, P.-Ä. Malmqvist, and B. O. Roos, J. Chem. Phys. 96, 1218 (1992).

B. O. Roos, K. Andersson, M. P. Fülscher, P.-Ä. Malmqvist, L. Serrano-Andrés, K. Pierloot, and M. Merchán, Adv. Chem. Phys. 93, 219 (1996). M. Merchán, L. Serrano-Andrés, M. P. Fülscher, and B. O. Roos, in

Recent Advances in Multireference Methods, edited by K. Hirao (World Scientific, Singapore, 1999).

39 L. Serrano-Andrés and M. Merchán, in Encyclopedia of Computational Chemistry, edited by P. v. R. Schleyer, P. R. Schreiner, H. F. Schaefer III, W. L. Jorgensen, W. Thiel, and R. C. Glen (Wiley, Chichester, 2004).

40 M. Merchán and L. Serrano-Andrés, in Computational Photochemistry, edited by M. Olivucci (Elsevier, Amsterdam, 2005), pp. 35-91.

41B. O. Roos, in Advances in Chemical Physics: Ab Initio Methods in Quantum Chemistry II, edited by K. P. Lawley (Wiley, Chinchester, 1987), Chap. 69, p. 399.

42S. Klein, E. Kochanski, A. Strich, and A. J. Sadlej, Theor. Chim. Acta 94, 75 (1996).

43P.-O. Widmark, P.-Ä. Malmqvist, and B. O. Roos, Theor. Chim. Acta 77, 291 (1990).

44N. Forsberg and P.-Ä. Malmqvist, Chem. Phys. Lett. 274, 196 (1997).

45 J. Finley, P.-Ä. Malmqvist, B. O. Roos, and L. Serrano-Andrés, Chem. Phys. Lett. 288, 299 (1998).

46 R. Crespo, M. Merchán, and J. Michl, J. Phys. Chem. A 104, 8593 (2000).

47P.-Ä. Malmqvist, Int. J. Quantum Chem. 30, 479 (1986).

48P.-Ä. Malmqvist and B. O. Roos, Chem. Phys. Lett. 155, 189 (1989).

49K. Andersson, M. Barysz, A. Bernhardsson, M. R. A. Blomberg, Y. Carissan, D. L. Cooper, M. Cossi, M. P. Fülscher, L. Gagliardi, C. de

Graaf, B. Hess, G. Hagberg, G. Karlström, R. Lindh, P.-Ä. Malmqvist, T. Nakajima, P. Neogrády, J. Olsen, J. Raab, B. O. Roos, U. Ryde, B. Schimmelpfennig, M. Schütz, L. Seijo, L. Serrano-Andrés, P. E. M. Siegbahn, J. Stálring, T. Thorsteinsson, V. Veryazov, and P.-O. Widmark, MOLCAS, version 6.0, Department of Theoretical Chemistry, Chemical Centre, University of Lund, Sweden, 2004.

50L. Karlsson, L. Mattson, R. Jadrny, R. G. Albridge, S. Pinchas, T. Bergmark, and K. Siegbahn, J. Chem. Phys. 62, 4745 (1975).

51 J. H. Fillion, R. van Harrevelt, J. Ruiz, M. Castillejo, A. H. Zanganeh, J. L. Lemaire, M. C. van Hemert, and F. Rostas, J. Phys. Chem. A 105, 11414 (2001).

52 G. Theodorakopoulos, I. D. Petsalakis, R. J. Buenker, and S. D. Peyer-imhoff, Chem. Phys. Lett. 105, 253 (1984).

53W. F. Chan, G. Cooper, and C. E. Brion, Chem. Phys. 178, 387 (1993).

54 C. Makochekanwa, R. Kajita, H. Kato, M. Kitajima, H. Cho, M. Kimura, and H. Tanaka, J. Chem. Phys. 122, 014314 (2005).

55 G. Theodorakopoulos, C. A. Nicolaides, R. J. Buenker, and S. D. Peyer-imhoff, Chem. Phys. Lett. 89, 164 (1982).

56 F. Schneider, F. Di Giacomo, and F. A. Gianturco, J. Chem. Phys. 104, 5153 (1996).

57 L. Serrano-Andrés, M. Merchán, I. Nebot-Gil, R. Lindh, and B. O. Roos, J. Chem. Phys. 98, 3151 (1993).