Ab initio MO–VB study of water dimer

Chemical Physics 232 Ž1998. 289–298

Ab initio MO–VB study of water dimer A. Famulari ) , M. Raimondi, M. Sironi, E. Gianinetti Dipartimento di Chimica Fisica ed Elettrochimica and Centro CNR - CSRSRC, UniÕersita` degli Studi di Milano, Õia Golgi 19, 20133, Milano, Italy Received 22 July 1997; in final form 19 March 1998

Abstract The equilibrium structure and binding energy of the water dimer system were determined by employing a general ab initio VB approach. Starting from the SCF–MI wavefunction, non-orthogonal virtual orbitals optimal for intermolecular correlation terms have been determined. BSSE is excluded in an a priori fashion and geometry relaxation effects are ˚ b s 134.58 and a s 2.58, in naturally taken into account. The equilibrium geometry corresponds to R O – O s 3.00 A, ˚ The estimated equilibrium agreement with the experimental values. The donor OH bond results elongated by 0.002 A. binding energy of the water dimer is y4.69 kcalrmol. Taking zero-point vibrational effects into account, the binding enthalpy is y3.1 kcalrmol, to be compared with the experimental estimate of y3.59 " 0.5 kcalrmol, determined from measurements of thermal conductivity of the vapour. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction The hydrogen bond plays a role of great significance in a vast number of chemical and biological phenomena. The prototype of the hydrogen bond is the interaction between two water molecules. The structure of the water dimer has been established by means of molecular beam electric resonance experiments w1,2x and by a notable quantity of theoretical ab initio studies. Very few measurements of the binding energy De have been performed. The most recent study is based on thermal conductibility of H 2 O and D 2 O vapours w3x. This method necessitates theoretical models to analyse the data. In fact, the measured association enthalpy, D H, of the dimer at a specified temperature includes the effect of thermal correction and of the zero-point energy ŽZPE.. It follows that the estimate of the experimental binding energy De requires the theoretical evaluation of this quantity. The value of y5.4 " 0.7 kcalrmol is obtained if a zero-point vibrational energy ŽZPVE. of 2.25 kcalrmol, computed at the harmonic approximation by Curtiss and Pople w4x, is included. A more accurate value for this quantity was estimated to be 2.43 kcalrmol w5x, but it was not BSSE corrected. In addition, since it was established that the intermolecular vibration modes are significantly anharmonic, a more reliable estimate of the ZPVE must require a dynamical calculation. In the case of the hydrogen fluoride dimer the harmonic approximation ZPVE, 1.9 kcalrmol, becomes 1.6 " 0.2 kcalrmol according to a dynamic calculation w6x. No

)

Corresponding author. E-mail: [email protected]

0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 1 2 2 - 0

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accurate ZPVE value is known for the water dimer, but there is consensus that it should be near 1.8 kcalrmol w7,8x. By employing such more accurate value, a new experimental value for De is y5.2 " 0.7 kcalrmol w9x. The theoretical variational supermolecule approach gives a De close to the experiment, but the calculations can be BSSE contaminated. The CP correction leads to a computed De between y4.5 and y4.7 kcalrmol. Frisch et al. w5x obtained a value of y4.56 kcalrmol by employing the many-body perturbation theory at the fourth order ŽMBPT4. with the frozen core approximation ŽFC. and a basis set of 150 elements indicated as 6-311 q q GŽ3df, 3dp.. Szalewicz et al. w10x have come to an estimate of y4.7 " 0.35 kcalrmol by employing the SAPT theory at the second order, as well as various supermolecule methods ŽMBPT4, coupled cluster with single and double excitations ŽCCSD., and CCSD to the fourth order with triple excitations ŽCCSD-T... Error bars reflect the estimates of basis incompleteness and of truncation error in perturbation theory. Hendriks and van Duijneveldt w11x came to an analogous result Žy4.7 " 0.3 kcalrmol. using MBPT2 and CEPA approximation. Rybak et al. w12x employed a higher-order SAPT and the MBPT4 approach, including over 100 basis functions and obtained a value of y4.7 " 0.2 kcalrmol. Feller w13x employed bases containing over 230 functions at the MBPT2 level. The CP corrected value obtained with the largest basis set, indicated as CC-pVQZ, was y4.67 kcalrmol. Van Duijneveldt et al. w14x performed an accurate analysis at the MBPT2 and CEPA levels. These authors accomplished a geometry optimisation, followed by CP correction at the MBPT2 level employing a basis set of 125 spdf terms including bond functions, and obtained an energy of y4.68 kcalrmol. Kim et al. w15x report a CP corrected value of y4.66 kcalrmol. Wang et al. w16x made use of a number of basis sets containing bond functions and found a CP corrected value of y4.75 kcalrmol. More recently, Mas and Szalewicz w9x obtained a value of y5.05 kcalrmol employing a higher-order SAPT with a basis set of 152 terms and bond functions. Oliveira and Dykstra w17x have questioned the use of bond functions as their use leads to a significant increase in the BSSE. The recent work of Feller et al. w18x reports a value of y4.72 " 0.1 kcalrmol at the MP2 level. In the present work a general molecular orbital–valence bond ŽMO–VB. ab initio approach to determine the water–water interaction energy will be described. The method represents the natural evolution of the MO–VB approach presented some time ago by Raimondi w19x. It makes use of a procedure which starts from the SCF–MI Žself-consistent field for molecular interaction. wavefunction w20x and specialises the algorithm, previously described by Raimondi et al. w21x in the context of the spin coupled theory, to generate virtual orbitals optimal for the treatment of dispersion forces.

2. Theory The general ab initio MO–VB wavefunction consists of single and double excitations out of the SCF–MI determinant to include intermolecular correlation effects. The theory involved will be presented briefly; a more detailed account can be found in Refs. w20,22x. The validity of the method extends from the long range to the region of the minimum and of short distances. It is based on the orbitals of the perturbed subsystems and in the long-range limit, turns out similar to a perturbation treatment. This enables the use of perturbation theory as a guide to gain insight for the choice of the configurations to be included and gives a physically based feeling for the interpretation of the results. The resulting wavefunction is size consistent; the elimination of BSSE is obtained in an a priori fashion, both at the Hartree–Fock and at the correlated level. In weakly interacting systems, such as van der Waals or hydrogen-bonded systems, the BSSE can be of the same order of magnitude of the interaction energy involved, and it represents therefore a major inconvenience. The counterpoise ŽCP. technique w23x is the most frequently employed a posteriori correction. Nevertheless, it has been underlined that the method introduced by Boys and Bernardi does not determine the BSSE magnitude precisely w24x as it does not become more reliable as the basis set size increases w25x. The SCF–MI ‘a priori’ strategy to avoid BSSE is always correct and not only for full CI wavefunctions as in the case of the CP a posteriori procedure w24x, which introduces other spurious contributions: the ‘secondary

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superposition error’. This improper modification of multipole moments and polarizabilities of the monomers w26x is particularly important in the case of anisotropic potentials, where the new errors in the wavefunction can alter the shape of the PES and the resulting physical picture. The algorithm for the SCF–MI procedure w20x has been recently implemented w27x into the GAMESS-US suite of programs w28,29x. According to this method, the supersystem AB formed by two interacting monomers A and B is described by the one-determinant wavefunction:

C 0 s Ž N !.

y1 r2

A F 1A Ž 1 . F 1A Ž 2 . . . . F NAA Ž 2 NA . F 1B Ž 2 NA q 1 . F 1B Ž 2 NA q 2 . . . . F NBBŽ 2 NA q 2 NB . .

Ž 1. The molecular orbitals of each fragment are left free to overlap with each other. The key of the method is the M partitioning of the total basis set x s  x k 4ks1 , so that orbitals of fragment A ŽF 1A . . . F NAA . and the orbitals of B B MA MB fragment B ŽF 1 . . . F N B . are expanded in two different subsets, x A s  x pA 4ps1 centred on A and x B s  x qB 4qs1 centred on B: MA

FaA s

x pA TpAa ,

a s 1, . . . , NA ,

Ž 2a .

Ý xqB TqBb ,

b s 1, . . . , NB ,

Ž 2b .

Ý ps1 MB

F bB s

qs1

where N s 2 NA q 2 NB is the total number of electrons and M s MA q M B is the basis set size. The procedure to exclude the BSSE in an a priori fashion is described in the companion paper in this issue w30x. A natural way to describe the intermolecular interaction, including the effects deriving from the overlap between the orbitals of the separated fragments, is the valence bond ŽVB. approach. Accordingly, a very compact multi-structure VB — non-orthogonal CI — calculation is carried out to determine intermolecular correlation effects. This scheme provides an accurate potential from which BSSE is excluded, while the orbitals are allowed to distort and polarise under the effect of the approaching partner. In order to reduce the VB expansion, a method was developed to obtain virtual orbitals corresponding to the SCF–MI wavefunction, which ensures an energetic contribution of the same accuracy as very extended CI calculations. To avoid the occurrence of BSSE also the virtual FaA) y F bB) SCF–MI orbitals are expanded only in the basis sets of their own fragment: MA

FaA s

Ý ps1

MB

x pA TpAa , F bB s

Ý xqB TqBb , qs1

MA A Fa) s

Ý ps1

MB

x pA TpAa) , F bB) s

Ý xqB TqBb ) .

Ž 3.

qs1

Such constraints imply the non-orthogonality of the orbitals. From the SCF–MI single-determinant wavefunction Ž1. it is possible to build excited configurations by substituting one or more of the occupied orbitals with virtual orbitals. To describe the effects of the dispersion contribution, vertical excitations consisting of two excitations localised on each fragment can be taken into account. A linear combination of the resulting configurations constitutes a multi configuration VB wavefunction. The energy of such wavefunction can be calculated by solving the corresponding secular problem, which includes the determination of the hamiltonian and overlap matrixes between non-orthogonal VB structures. It is possible to realise this procedure employing a general VB code w31x. In order to describe accurately the interaction between two fragments, a sufficient number of virtual orbitals should be included into the calculation. This poses a limit in the calculations relative to systems with a great number of electrons, because of the number of configurations which must be considered. As a consequence, it is necessary to possess a scheme to obtain a set of optimal virtual orbitals, in order to generate a compact VB wavefunction.

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292

A The optimal virtual orbitals Fa) and F bB) are determined according to the approximation that they maximise separately the dispersion contribution of each of the two configuration wavefunctions b) C X s C0C 0 q C a bCaa) , b

Ž 4.

b) where Caa) represents a doubly excited configuration in which electrons are excited from the occupied b A SCF–MI orbitals FaA and F bB to the virtual orbitals Fa) and F bB) , respectively. A B The associated virtual orbitals Fa) and F b ) are determined at the variation–perturbation level of theory by minimisation of the second-order expression of the energy:

HŽ0, a b. y H00 SŽ0 , a b.

Ž2.

E s H00 q

2

H00 SŽ a b , a b. y HŽ a b , a b.

,

Ž 5.

where H00 s ²C 0 < H
b) : SŽ0 , a b. s ²C 0
b) : HŽ0, a b. s ²C 0 < H
b) < b) : SŽ a b , a b. s ²Caa) Caa) , b b

b) < b) : HŽ a b , a b. s ²Caa) H
by satisfying the conditions EE Ž2. A EFa)

EE Ž2. s0 ,

EF bB)

s0 .

Ž 6.

A The energy E given by Eq. Ž5. is a function of the virtual orbitals Fa) and F bB) through the elements of the overlap matrix and of the hamiltonian operator. Such matrix elements, expressed in terms of wavefunctions containing non-orthogonal orbitals, can be evaluated by Lowdin formulae w32x. To determine the virtual orbitals ¨ which minimise E, the derivatives of the expression Ž5. with respect to the coefficients of the expansion Ž3. is considered:

EE ETmAa)

s AŽ a b.

EHŽ0, a b. ETmAa

y H00

ESŽ0 , a b. ETmAa

q BŽ a b.

EHŽ a b , ab. ETmAa

y H00

ESŽ a b , a b. ETmAa

,

Ž 7.

where AŽ a b. s

2 Ž HŽ0, a b. y H00 SŽ0 , a b. . H00 SŽ a b , a b. y HŽ a b , a b.

,

BŽ a b. s

HŽ0 , a b. y H00 SŽ0 , a b. H00 SŽ a b , a b. y Ha b , a b

2

.

Ž 8.

The strategy that we have employed has been described in w33,34x and it is based on a ‘Laplace expansion’ of supercofactor elements. By performing several algebraic transformations elsewhere reported w21x, it is possible to demonstrate that the conditions reported in the relations Ž6. lead to the solution of the following set of equations

½

FaA TaA s y´ aA ,

a s 1, 2, . . . , NA

FbB TbB s y´ bB

b s 1, 2, . . . , NB

,

.

Ž 9.

The minimisation strategy is the specialisation to the case of a SCF–MI zero-order wavefunction of the algorithm previously described by Raimondi et al. w21x in the context of the spin coupled theory. Acceleration of

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293

the minimisation procedure now includes the approximate evaluation of the hessian as described by Clarke et al. w35x. It should be underlined that in this scheme the a priori exclusion of BSSE is still operative, as the virtual orbitals are expanded on the two distinct subsets centred on A and B, see Eqs. Ž3.. Starting from a guess, it is possible to solve Eqs. Ž9. iteratively for a new set of TaA and TbB , until convergence is reached. An algorithm to perform this procedure has been implemented; convergence and stability reveal to be satisfactory. The virtual orbitals obtained by means of the perturbation theory approximation are then employed to construct a full set of all the singly- and doubly-excited configurations which provide a final VB-like wavefunction. The final wavefunction is assumed to have a general MO–VB form: Na

C s C0C 0 q

Nb

Na

Nb

b) q Ý C aCaa) q Ý CbC bb ) Ý Ý Ca bCaa) b

as1 bs1

as1

Ž 10 .

bs1

and represents the configuration interaction between the SCF–MI wavefunction

C 0 s
Ž 11 .

the singly excited localised configuration state functions A Caa) s
Ž 12a.

C bb ) s
Ž 12b.

and the doubly excited localised configuration state functions b) A Caa) s
Ž 13 .

obtained by simultaneous single excitation localised on A and B. The singlet spin function for the two or four electrons involved in the single or double excitation are: 4 Q 0,2 0; 1 and Q 00 s C1Q 0,4 0; 1 q C2 Q 0,4 0; 2 .

Ž 14 .

The configurations included in the VB wavefunction play a significant role in the field of intermolecular forces, as they can be associated with specific physical effects. By writing the wavefunction obtained from a SCF–MI calculation as C 0 , one can use it as a building block to construct various VB structures, which, asymptotically, in absence of relaxation effects, would coincide with specific perturbation contributions: C 0 b) coulombian or exchange energy; Caa) and C bb ) polarisation or induction energy; Caa) dispersion energy. b It has to be underlined that the last doubly excited structure is optimal to describe the correlation among the electrons of different fragments. The single vertical excitations have been added in order to refine the relaxation of the occupied SCF–MI orbitals, which were kept fixed during the virtual orbital determination procedure. Finally, the multi structure VB Žnon-orthogonal CI. problem is set up and solved variationally according to standard VB techniques, see Raimondi et al. w31x and Cooper et al. w36x.

3. Results and discussion The water dimer interaction energy was investigated at the Hartree–Fock and correlated level using the method described in the previous section.

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A. Famulari et al.r Chemical Physics 232 (1998) 289–298

Table 1 Basis set Žsee Ref. w37x. Atomic orbital

Exponent

Contraction coefficient

105375.000 15679.200 3534.540 987.365 315.979 111.654

0.000046 0.000361 0.001920 0.008206 0.029725 0.090452

Oxygen: 1s

2s

42.2875 17.3956

0.217402 0.368720

3s

7.43831

1.0

4s

3.22286

1.0

5s

1.25388

1.0

6s

0.495155

1.0

7s

0.191665

1.0

8p

200.000000 46.53340 14.62180 5.31306

0.000892 0.007351 0.034863 0.114877

9p

2.102520 0.850223

0.255983 0.374236

10p

0.337597

1.0

11p

0.128892

1.0

12d

1.0

1.0

13d

0.3

1.0

Hydrogen: 1s

271.6048051 40.7182982 9.2517552 2.6136590 0.8511307

0.001699549 0.013344571 0.066581390 0.260894100 0.738689100

2s

0.3054946

1.0

3s

0.1150430

1.0

4s

0.05

1.0

5p

1.4

1.0

6p

0.3

1.0

A. Famulari et al.r Chemical Physics 232 (1998) 289–298

295

Table 2 Comparison of the present and previous works energy results Oxygen

Hydrogen

N

Emo n Žhartree.

D ESCF Žkcalrmol.

D ESCF – MI Žkcalrmol.

Ref.

20s 10p 6d 6f 7s 4p 2d a

10s 6p 6d 4s 2p a

548 102

y76.06760761 y76.06424166

y3.71 y3.84

y3.33 y3.32

w30x present work

a

Ref. w37x.

Table 3 Optimised geometrical parameters for the water dimer system Žsee Fig. 1. compared with some values reported in literature

˚. R O – O ŽA b Ž8. F Ž8.

SCF–MI present work

VB present work

Exp. Ref. w1x

SAPT MBPT4 Ref. w9x

MP4 Ref. w14x

3.16 136.7 2.9

3.00 134.5 2.5

2.98"0.03 122.0"10 0.0"10

2.953 124 6.8

2.949 124.8 5.35

The w7s 4p 2d r 4s 2px basis set proposed by Millot and Stone w37x was employed Žsee Table 1.. The oxygen atom is described by a 7s 4p 2d set given by Hess w38x, and for the hydrogen atom a 4s 2p set is used, which is the result of a contraction and scaling of a basis set proposed by van Duijneveldt w39x and contains one s and two specifically optimised p functions. This basis set gives a monomer energy close to the Hartree–Fock limit Žy76.076 au.. It also reproduces well the properties of the water dimer system as obtained in the companion paper in this issue Žsee Table 2. using the basis set of 548 functions. Optimal geometrical parameters are also comparable to those obtained with this very large basis set Žsee Table 3 and Fig. 1.. In order to take intermolecular correlation effects into account, the VB wavefunction described above was employed. For each water molecule we considered an actiÕe space of four MOs with one MO Žoxygen 1s 2 electrons. kept frozen. Different optimised virtual orbitals are determined by considering each double excitation involving an electron of group A and an electron of group B, respectively. The number of virtual orbitals obtained for each occupied MO is equal to the number of occupied MO on the other fragment. The resulting dimension of the Õirtual space of each fragment is 16, corresponding to four virtual orbitals for each occupied MO. This implies a set of 32 Õertical singly excited configurations. By adding all the corresponding 256 Õertical doubly excited spatial configurations the final VB-like wavefunction results of 289 configurations. By taking the dimension of the spin space into account, the size of the resulting VB matrix is 545.

Fig. 1. Intermolecular coordinates in the water dimer system.

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Table 4 Calculated and experimental water dimer binding energy

Ref. w37x

D E VB Žkcalrmol. present work

D E Exp Žkcalrmol. Ref. w9x

D E Exp Žkcalrmol. Ref. w3x

4s 2p

y4.69

y5.2"0.7

y5.44"0.7

Oxygen

Hydrogen

Ref. w37x 7s 4p 2d

Table 5 Estimated and experimental water dimer binding enthalpy SCFMI ZPE Žkcalrmol. present work

Estimated ZPE Žkcalrmol. Ref. w9x

VB enthalpy Žkcalrmol. present work

EXP enthalpy Žkcalrmol. Ref. w3x

1.60

1.6r1.8

y3.1

y3.59"0.5

The resulting optimal geometry turns out in good accordance with the experimental data Žsee Table 3 and Fig. 1.. As expected, the inclusion of correlation leads to an intermolecular separation that is shorter than the SCF–MI predicted value. The geometry of the acceptor molecule remains equal to that of the isolated monomer, ˚ and the other O–H lengthens by while the hydrogen bonded O–H of the donor shortens by 0.002 A approximately the same amount. The corresponding De of y4.69 kcalrmol falls within the experimental range Žsee Table 4.. To compare the calculated binding energy De with the experimental dimerization enthalpy Ž D H . it is necessary to consider the thermal correction D Ethermal that includes vibrational Ž D Evib ., rotational Ž D Erot . and translational Ž D Etransl . energy contributions. Following Curtiss et al. w3x the experimental dimerization enthalpy can be expressed as D H 373 s De q D Ethermal , where D Ethermal s D Evib Ž T . q D Erot Ž T . q D Etransl Ž T . q D Ž PV . s D Evib Ž T . y 4 RT . An early estimate of D Ethermal at 373 K was q1.85 kcalrmol w3x, but values in the range 1.7–1.92 kcalrmol have been also reported w40–42x. Very recently w9x proposed a D Evib of 1.8 kcalrmol and a corresponding D Ethermal contribution of 1.6 kcalrmol. Following the suggestion of Mas and Szalewicz w9x we have performed full Cartesian SCF–MI geometry optimisation to include the variation of both intermonomer and intramonomer coordinates. At the SCF–MI level of theory and employing Millot basis w37x our D Evib is 1.77 kcalrmol. This result was confirmed using a more expanded basis set w20s 10p 4d r 10s 4px consisting of 236 functions. By taking into account the temperature-dependent part of D Ethermal estimated at the harmonic level w18,40–42x, we found a final value for D Ethermal of 1.47 kcalrmol and a corresponding dimerisation enthalpy of y3.2 kcalrmol to be compared with the experimental value of y3.59 " 0.5 kcalrmol w3x. By employing the thermal contribution suggested in Ref. w9x the estimated enthalpy becomes y3.1 kcalrmol, a result which remains in sufficient agreement with the experiment ŽTable 5..

4. Conclusions The results confirm that the VB-like wavefunction based on the SCF–MI non-orthogonal occupied and virtual orbitals describes accurately the intermolecular potential of the water dimer. The estimated contribution

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of dispersion is 1.36 kcalrmol and corresponds to slightly less than the 30% of the total De . BSSE is excluded in an a priori fashion and geometry relaxation effects are naturally taken into account. ˚ b s 134.58 and a s 2.58. is in good agreement with The calculated equilibrium geometry Ž R O – O s 3.00 A, the experimental values w1x. The geometry of the acceptor remains unchanged, while the donor OH bonds distort ˚ The estimated De of the dimer is y4.69 kcalrmol. Taking D Ethermal contributions into account, the by 0.002 A. computed binding enthalpy turns out to be in sufficient agreement with the experimental value determined by measurement of the thermal conductivity of vapour. The basis set employed reproduces the computed properties obtained in earlier work on near-Hartree–Fock limit properties of the water dimer. The SCF–MI geometrical parameters are also reproduced. Using the same VB-like approach a new ab initio potential was successfully used for molecular dynamics simulation of the liquid phase at room temperature w43,44x. By means of extensive calculations on the dimer and on the trimer a new NCC-like potential w45x was determined. The results showed good accordance with the experimental data relative to radial distribution functions, thermodynamic properties and geometric parameters. The computed IR and Raman frequencies shifts also agreed with the experimental values. The prediction of the internal pressure shows a great improvement in the computed value and an increased stability. In addition, the procedure correctly describes the physical behaviour of water at the critical point, where a fundamental variation in the hydrogen bond is observed w46x. The aim of the present work was to perform a first test of the reliability of the general VB approach to the study of the hydrogen bond. The complete elimination of BSSE in an a priori fashion and the absence of orthogonality constraints seems of vital importance to generate an ab initio potential which correctly reproduces the anisotropy of the intermolecular interaction. The present study will be completed employing a more extended basis set for a rigorous determination of the contribution of dispersion forces. In addition, we intend to study intramolecular correlation effects by adding to the MO–VB wavefunction employed in this work other classes of excitations constructed by means of a new set of specifically optimised virtual orbitals following a strategy already applied in the context of a multiconfiguration spin coupled approach w21,35x.

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