A correlated bender approach to non-linear molecules

Physica 94C (1978) 113-124 0 North-Holland Publishing Company

A CORRELATED BENDER APPRbACH TO NON-LINEAR MOLECULES” D. ROGOVIN Science Applications, Inc., La Jolla, Gdifomia 92037, USA Received 20 May 1977 Revised 20 September 1977

We present a theory of the bending mode vibration of non-linear.molecules that simulates correlations between the bond angle and bond length fluctuations. As a specific application we calculate the roto-vibration structure of water in the ground and first excited bending mode states. We demonstrate that the correlations discussed here significantly affect the structure of these vibrational bands

1. Introduction

In section 2 we present our two models of water vapor and discuss their physical characteristics. Specifically, we begin with the Darling-Dennison hamiltonian for a polyatomic molecule [ 11 and in a series of steps reduce it to two different forms that are appropriate for treating our two models of water. Following this we present a detailed description of the vibrational structure of Hz0 that our two models predict. Specifically, we discuss the potential energy surface, the characteristic structure of the protonproton motion as well as the rotation-constant matrices associated with these two models. In section 3 we consider the rotational level structure within the ground and (010) vibrational bands. Specifically, we obtain the rota-vibrational energies and wave functions using a modified theory of the asymmetric rotor. In particular, we diagonalize the roto-vibrational energies together, not individually as is usually done. This combined diagonalization is particularly important for H20 inasmuch as the rotation-vibration interaction is extremely strong and consequently plays an important role in the molecular energy level structure. We present tables of predicted versus experimental values of the various roto-vibration energy levels of H20. To determine how much of the rotation-vibration interaction is accounted for in our models, we also include in these tables the values of the rigid rotor energies. These were obtained by replacing all off-diagonal elements of the rotation matrices with zero [2]. A comparison of the various energies demonstrates that for the lOI0 state, about

In this paper we present a model for treating the roto-vibrational structure and spectra that arises from bond angle fluctuations in molecular systems and apply it to the specific case of water. Our model is based on the motion of the bond angle and the correlation of this motion with the stretch modes. Furthermore, our approach involves the description of a single internuclear coordinate and, as we shah explicitly demonstrate, is vastly superior to a treatment based on bond angle fluctuations alone. Specifically, we construct two models of the water molecule that are based on the relative motion of the two protons, with respect to each other, and the coup ling of this motion to the molecular rotation. In the first model, which is presented for the purposes of comparison, we treat only the fluctuations of the bond angle, so that the classical trajectory of the two protons is constrained to lie along the arc of a circle whose radius is equal to the average value of the OH bond length. We refer to this as the rigid bender model and note that it should be regarded as a standard approach to the rotation-vibration structure. In the second model the corresponding classical trajectory of the two protons is a straight path so that correlations between the OH bond length and the bond angle fluctuations are included. We shah refer to this as the correlated bender model. *Supported by Kirtland Air Force Base. 113

D. Rogovin/A correlated bender approach to non-linear molecules

114

9076 of the rotation-vibrbtion interaction is accounted for in the correlated bender model whereas the rigid bender accounts for only 80% of this energy. In section 4 we discuss the symmetric isotopic molecules D,O and T,O. This enables us to examine certain features of the cofrelated bender model that are mass-dependent. Spe ‘fically, the degree of centrifugal distortion, the molecular structure and the 7 correlation between the bond angle fluctuations and the OH stretch motion. Finally, in section 5 we present our conclusions on the correlated bender approach and compare our model of water to others in treating this unique molecule [1,3-111.

k

where U is given by

2. Foundations for quantum analysis In this section we present two models of the water molecule that are based on the fluctuations of a single coordinate. For convenience, we have subdivided this section into two parts. In section 2.1 we construct the two quantum mechanical hamiltonians that we use to model the rotation-vibration structure and spectrum of the water molecule. In section 2.2 we use these hamiltonians to derive various properties of Hz0 that depend on the intranuclear motion, and in addition present the potential energy surfaces. The properties of these two models are compared and their differences discussed.

2.1. Rotation-vibration

A, is the oath component of the vibrational-angular momentum. /.A~@ is the (CY,fi) component of the effective reciprocal moment of inertia tensor, and Mis its determinant. These quantities are determined by the molecular configuration and therefore are functions of the normal coordinates. Watson [ 121 has demonstrated that for non-linear molecules the Darling-Dennison hamiltonian can be cast in the following very useful form:

(2.3) a

It follows from eq. (2.3) that U depends only on the nuclear masses, the molecular geometry and the normal coordinates. Hence, it is independent of the various momenta that appear in eq. (2.1) and, as noted by Watson, can be regarded as an effective mass dependent potential which we formally include in V. We note that, when a molecule assumes a linear configuration, the nuclear moment of inertia associated with that direction vanishes and as a consequence, u+

hamiltonian

Our starting point is the Darling-Dennison [ 1] hamiltonian for a non-linear polyatomic molecule

-=.

(2.4)

The presence of an infinitely attractive term in a molecular hamiltonian is disturbing, and on physical grounds should not be present. The resolution of this difficulty can be obtained by noting that for linear molecules the rotation-vibration hamiltonian can be written as [13]

(2.1) k

In eq. (2.1) V is the molecular potential energy surface which is a function of all the normal coordinates @&= I,* . .}, Pk the corresponding conjugate momenta, IIor is the cwthcomponent of the total rotational-angular momentum of the molecule, and

+fCp2k+v,

(2.5)

k

where the quantities & and &Y,, are defined in ref. 13 and do not directly concern us here. The important

D. Rogovin/A correlated bender approach to non-linear molecules

feature of eq. (2.5) is the absence of the effective potential U. We note that eqs. (2.2) and (2.5) differ considerably in form, particularly in regard to the rotational structure [II, is not present in (2.5)] and the fact that linear molecules have an additional vibrational degree with respect to non-linear molecules. These features are well known; our only purpose in discussing them is to point out that the molecular hamiltonian is well behaved in the sense that it does not contain terms that diverge to minus infinity [14]. In the remainder of this paper we shall group the effective potential U with the actual potential V and refer to the sum, U + V, as V. It is this quantity that will be obtained by a least-squares fitting technique. We note that as a consequence the potential energy that we eventually obtain is mass dependent. We will investigate this mass dependence by examining the isotopic species D,O and T,O in section 4. For a symmetrical molecule such as H20, it is easy to demonstrate that the reciprocal moment of inertia tensor is diagonal, thus the hamiltonian reduces to

k

a

Furthermore, we shall neglect both Corlolis forces as well as the vibrational-angular momentum. This approximation is justified on the grounds that the bulk of the rotation-vibration interaction in the (0, 0,O) and (0, 1,O) bands arises from centrifugal distortion. Thus, our molecular hamiltonian reduces to (2.7) a

k

r(e) = &.&rO

+ nzH(1 + cos e)] /M&O,

energy that arises from fluctuations in 1(e) amounts to less than 0.08%. Now, the Born-Oppenheimer theorem [ 15] states that the coupling between the electron and nuclear motion is on the order of (m$M)t where M is some suitable nuclear mass. Thus, errors encountered in neglecting fluctuations of I(e) appear to be of the same order of magnitude as corrections that arise from deviations from an adiabatic model. Furthermore, as we shall demonstrate in section 4, the corresponding energy correction is an order of magnitude smaller than any other contribution that arises from approximations made in this paper. Consequently, we shall ignore the angular dependence of Z(e) and use for the kinetic energy of the bending mode +i2 a2 T=_-21(eo)ae2 ’ where 10 is the moment of inertia evaluated at a fixed angle 8, (see section 2.2). Next, we note that for situations involving large amplitude changes in the bond angle, as occurs in water, it is more convenient to work with a length than an angle. Thus, we define x =

2r

sin

e/2,

+i2

(2.10)

0

Fixing the OH bond length at r, we obtain the following hamiltonian

nt

Xl = t TE- + T, + V&I). (2.8)

where r is the OH bond length, mH, m. and MHz0 the atomic hydrogen, oxygen and molecular water masses, respectively. We note that I(e) depends only weakly on the bond angle 0 and in section 2.2 we demonstrate that the contribution to the vibrational

(2.9)

which varies from zero to 2r as 0 vanes from zero to A. Defining the dimensionless coordinate (z as x/r, we find that the kinetic energy associated with fluctuations of the bond angle is

Tl =r

Next, we consider the kinetic energy associated with the bending mode vibration, i.e. fluctuations in the bond angle 8. We begin by noting that the moment of inertia I(e) associated with the bending mode is

115

(2.11)

WY

where V,(q) will be taken to be a power series of the form V,(4) = 2 n=2

a?’ k - qo]!

(2.12)

D. Rogovin/A correlated bender approach to non-linear molecules

116

The parameters 4. = 2 sin eo/2, I and ai’) are chosen so that the eigenvahres and eigenvectors of H, will least-squares fit the A and B rotation constants of the (000) and (010) vibrational states as well as the spacings between the ground and the first three excited bending mode states. This completely defines the first model. In our second model we wish to constrain the classical trajectory in such a way that the two protons will travel in straight paths. Now, the variation of the kinetic energy operator with 4 as well as the presence of the first derivative term q(d/dq) in eq. (2.10) reflect the fact that the classical trajectory of the two protons is along the arc of a circle of radius r. Thus, if we neglect the first derivative term and fix 4 to a value 4i21 (discussed below), then classically we are describing the motion of two particles of mass ~1 (see section 2.2):

“zo+ mHi1 -(4&2’/2)21 = 2a3g8 “H Cc’mH [i -(4f)/2)2]MH,

(2.13)

traveling along a straight path. The molecular hamiltonian X2 for this model is

xc,=+;g-f$

$+

V2(4),

(2.14)

where v,(4) = 2

eL2’(4 - 462?

(2.15)

n--2

The parameters 4b2), r and ai2) are chosen in precisely the same way as in the first model, although, as we shall see, except for r their values are somewhat different than in the rigid bender model. 2.2. Vibration structure of the water molecule In this section we present the vibrational properties of the water molecule that are predicted by our two models. Specifically, we display tables of (1) the potential energy constants, (2) expectation values of various moments of the vibrational coordinate, (3) the A, B and C rotation constant matrices, and (4) the geometric structure of the molecule. For convenience, these properties are displayed in separate sections.

Table I Potential energy constants in cm‘-’ Parameter a2 a3 04

a5 a6 a7

aa

Rigid bender

Correlated bender

46 325 6 310 1969 113 149 118 476 -4 864 10 886

45 510 0 -30 833 0 -23 665 0 1820

Table II Wavenumber deviations of transitions

6Wl 6w2

6w3

Rigid bender

Correlated bender

3.260 -1.645 0.285

-0.928 0.661 -0.124

2.2.1. Potential energy function In table I we have displayed the various potential energy constants for the two models. An examination of this table reveals that the harmonic portions of the two potentials deviate by nearly 2% and the anharmanic pieces differ considerably. Furthermore, the correlated bender contains only even powers of (4 - 40) and we comment on this below. We note that including higher order terms does not significantly improve the fit. At first glance, one might suppose ,that a power series expansion in (4 - (lo) would be sufficient to fit three level spacings. However, our scheme requires that the A and B rotation constants be accounted for as well, and this apparently affects the fitting to the vibrational energy level structure. This statement is underscored by the fact that the greatest deviation in both models appears in the ground to first transition frequency (table II). Finally, we note that by choosing different weighing schemes we can somewhat improve the rigid bender’s fit to the first vibrational spacings. However, upon carrying this out we find no significant changes and the overall description of the molecule does not improve [ 171. 2.2.2. Proton-proton motion Next, we examine the ground state expectation values of the even moments of (4 - 40). An examination of table III reveals that the lower-order even

D. Rogovin/A correlated bender approach to non-linear molecules Table III Ground state even moments of (4 - 40) Moment

Rigid bender

Correlated bender

(4 - so)2 (4 - q(+4 (4 - 40);

0.911 0.248 0.112 0.669 0.551

0.910 0.250 0.114 0.721 0.578

2 IF;10 0

x x x x x

10-Z 10-S 10-4 lo+ 10-7

x x x x x

10’ 10-S lo4 10d lo-’

117

(0.01% deviation) whereas the first off-diagonal A element deviates by nearly 7% (0.4 cm-l). As we shah see, this feature is strikingly reflected in the rotation-vibration energies. A further examination of tables IV reveals that, although the ground state A and B rotation constants are fit equally well by our two models, the predicted Table IV(a) Rotation constant matrix from the rigid bender model

moments of the two models are virtually identical whereas the higher ones deviate by 5%. We also find that all the odd moments of the rigid bender are virtually zero, e.g. (4 - qo) 9 10”. Thus, the net result of the odd powers of the rigid bender potential is to nullify the term q(d/Q) s qO(d/dq) in the kinetic energy. This explains the lack of odd terms in the correlated bender’s potential, V,(4). We note that the mean distance between the two protons remains unchanged by zero point fluctuations (i.e. (OIqIO> = qo), although the bond angle 19= 2 sin-l 412 does deviate from its equilibrium value (see below). Since the various moments measure the mean distance, etc. of the two protons it follows that deviations between the rigid bender and correlated bender are greatest when 14- 401 is large, i.e. when the molecule is near the linear configuration. 2.2.3. Rot&ion constants It is well known that the rotational level structure within a given band is determined by the rotation constant matrix. In tables IV(a-c) we have displayed the A, B and C rotation constant matrices for the rigid bender, correlated bender and the experimentally deduced values in units of wavenumbers [ 191. An examination of tables IV(a) and (b) reveals that although the diagonal elements of the two models are essentially the same the off-diagonal elements differ considerably. For example, the ground state A rotation constant differs by 0.003 cm-l in the two models

V

A rotation constants

0 1 2

27.873 -5.959 1.317

-5.959 31.743 -10.953

B rotation constants 0 1 2

14.510 1.733 0.449

1.733 14.475 2.514

0.449 2.5 14 14.400

0.082 9.206 0.192

0.013 0.192 8.981

C rotation constants 0 1 2

9.315 0.082 0.013

Table IV(b) Rotation constant matrix for the correlated bender V

A rotation constants

0 1 2

27.876 -6.358 2.504

-6.358 31.470 -11.167

2.504 -11.167 36.394

B rotation constants 0 1 2

14.510 1.781 0.227

1.781 14.837 2.632

0.227 2.632 15.136

0.084 9.284 0.143

-0.009 0.143 9.214

C rotation constants 0 1 2

9.297 0.084 -0.009

Table IV(c) Experimental rotation constants V

A rotation constant

B ro tatton constant

C rotation constant

0 1

27.876 -

14.512 -

9.285 -

31.12

1.317 -10.953 37.649

14.66

9.15

D. Rogovin/A

118

correlated bender approach to non-linear molecules

value of the ground state C rotation constant is slightly better in the second model. We note that deviations from experiment for this rotation constant are 0.32 and 0.13%, respectively. Next, we note that the correlated bender presents a somewhat more reliable description of the first excited state rotation constants than does the rigid bender. A very interesting feature of this state is that both the A and B rotation constants increase relative to the ground state. Now, on naive grounds one might expect that as the molecule opens up (i.e. the bond angle increases) the B-rotation constant should decrease. The fact that it does not is indicative of a correlation between the ground state OH stretch mode and the first excited bending mode. This interpretation is underscored by our calculation. Next, we note that the strength of the rotationvibration interaction is set by the size of the offdiagonal matrix elements of the rotation constants. For example, from the correlated bender model we have for the A-rotation constant (OlAll)

I

I o.22; I

I

(s


= 0.36,

the correlated bender yields a static bond angle that lies within 0.05% of the experimental value. Furthermore, the equilibrium bond angle is within 0.09% of the measured value. This should be compared to the distorted bender values of 0.35 and 0.24%. Note that the bond length parameter T is the same in both models, which implies that the correlations between the stretch modes and the bond angle fluctuations that are included in our second model of Hz0 affect only the bending mode wave functions. We can also demonstrate that fluctuations of the moment of inertia, I(e), associated with the bending motion are not significant. The size of this effect is set by the ratio

This corresponds to an energy of 0.32 cm-l (the ground state energy of HZ0 is about 850 cm-l) and does not represent a significant effect in the rotationvibration structure of the molecule. Finally, the hydrogen-hydrogen separation for the correlated bender is 1.53 A.

and for the B constant, 3. Rotational-vibrational

(WI 1)_ o COlBlO> ’

(llBl2) =

12.

m

’

o

structure of water

18

**

Note the dramatic increase in these ratios as we go from the ground to the first excited state which is to be expected as the molecule opens up and becomes less rigid. 2.2.4. Molecular geometry In table V we present predicted versus measured values for the bond length, bond angle and B0 for the two models. An examination of the data reveals that

As we discussed in sections 1 and 2, the rotationvibration structure of the water molecule is dominated by centrifugal distortion. Since our models directly incorporate this feature an excellent test of their accuracy is the rotation-vibration energy levels themselves. In this section we present theoretical calculations through J = 10 of the ground and first excited bending mode states using our two models of water. In general, we find that the correlated bender displays a much more reliable picture of water as its predicted energy levels lie closest to experimental

Table V Geometric structure of water Parameter

Observed [8-lo]

Rigid bender

Dev. (So)

Correlated bender

Dev. (%I

r

0.972 A 104.50° 103.9O

0.969 A 104.7S0 104.27”

0.31 0.24 0.35

0.969 A 104.41” 103.95”

0.31 0.09 0.05

(8) 00

119

D. Rogovin/A correlated bender approach to non-linear molecules

findings. Furthermore, to measure the significance of the off-diagonal rotation constant matrix elements, we also present predicted rotational energies for a model in which these matrix elements have been set equal to zero. We note that these matrix elements represent the coupling between the various vibrational states and are a measure of the importance of centrifugal distortion on the structure of the molecule. In table VI the predicted rotational-vibrational level structure in the ground vibrational states through J = 10 are given. As we are specifically interested in the effects of centrifugal distortion on the structure of the molecule, we present only those states which have r > 0. It is well known that such states involve rotation about the small momenta of inertia (A and B). On physical grounds it is clear that Table VI Comparison of rotation-vibration

energy levels (in cm-‘) in the (0,0, 0) state

Observed

11

20 22 30 33 40 42 44 50 53 55 60

62 64 66 70 73 75 71 80 84 80 90 95 97 99 100 104 lo6 lo8 1010

42.31 95.18 136.16 206.30 285.41 815.78 383.84 488.13 503.97 610.34 742.01 661.55 757.78 888.63 1045.06 921.75 1059.83 1216.19 1394.81 1131.77 1411.83 1789.12 1476.99 1810.59 2010.0 2225.55 1724.71 2054.37 2254.36 2471.59 2702.09

the coupling between molecular rotation and vibration is strongest for these states. This feature is reflected in the predicted level structure. Specifically, an examination of table VI reveals that deviations between the model without centrifugal distortion and experiment increases drastically with increasing rotational energy. For example, the 9, and 10, states deviate from the experimentally derived values by only 1.09 and 1.32%, whereas for the 99 and 10, states the deviation is 6.32 and 7.62%. Next, we consider the predicted rotationvibration spectrum of the rigid bender. An examination of table VI reveals that the predicted levels all he within 1.5% of the experimental values. Furthermore, through J = 8, all but one level lies within 1% of the correct values. Next, we determine how well this

Rigid rotor

42.38 95.20 136.53 206.69 287.22 316.45 385.79 494.49 506.17 616.49 757.39

664.80 764.17 903.43 1076.03 935.07 1074.59 1246.28 1450.41 1141.54 1441.61 1880.52 1493.23 1864.32 2099.19 2366.37 1747.54 2109.03 2342.81 2609.26 2907.95

Rigid bender

Dev. (%)

Correlated bender

Dev. (%)

92.22 136.26 206.61 285.90 316.23 384.73 489.57 505.44 612.62 745.43 663.49 761.08 893.49 1051.62 932.33 1066.29 1225.25 1406.43 1137.49 1423.52 1807.56 1486.09 1830.13 2033.89 2253.71 1738.09 2078.64 2284.36 2506.95 2742.40

0.00 0.04 0.07 0.15 0.17 0.14 0.23 0.30 0.29 0.37 0.45 0.29 0.44 0.54 0.63 0.49 0.61 0.14 0.83 0.51 0.84 1.03 0.62 1.08 1.19 1.27 0.78 1.18 1.33 1.43 1.49

42.31 95.21 136.18 206.48 285.52 316.14 384.36 488.48 505.02 611.58 743.04 663.14 760.08 891.24 1047.30 931.30 1064.10 1221.05 1399.37 1136.57 1419.50 1797.40 1484.00 1823.60 2023.88 2239.60 1736.16 2072.37 2274.71 2493.26 2724.24

0.00 0.03 0.01 0.09 0.04 0.11 0.14 0.07 0.21 0.20 0.13 0.24 0.30 0.29 0.21 0.38 0.40 0.40 0.32 0.42 0.56 0.46 0.47 0.12 0.69 0.63 0.66 0.88 0.90 0.88 0.82

D. Rogovin/A correlated bender approach to non-linear molecules

120

model accounts for centrifugal distortion. We note that for the 8, state centrifugal distortion amounts to 91.4 cm-l (i.e. about 5% of the rotational energy) and the rigid bender accounts for 72.96 cm-l or 79.8% of these effects. For lOlo, where the centrifugal distortion amounts to 205.86 cm-l, the rigid bender accounts for 165.55 cm-l or 80.4%. Thus, summarizing the results of table VI for the rigid bender we can state that this model accounts for about 80% of the effects of rotational distortion on the structure of the molecule. Finally, we note that all deviations are systematic, i.e. of the same sign, so that it is unlikely that there is any cancellation of errors. Next we consider the predicted level structure of the correlated bender. We first note that all values throughJ = 10 lie well within 1% of the experimental values. As some of these lie above the (010) J = 0 vibrational state, it is clear that this model presents an accurate picture of the structure of the molecule. To obtain a more detailed account of the model’s success, we again examine centrifugal distortion. For the 8, state the correlated bender accounts for 83.12 cm-l or 90.9% of the effects of centrifugal distortion. Similarly, for the lOlo state it accounts for 183.71 cm-l or 89.2% of the energy. Thus, the correlated bender accounts for about 90% of the effects of centrifugal distortion in H20. In table VII we summarize these considerations for the important high energy states. An examination of table VII reveals that the corre-

lated bender accounts for 85-90% of the energy due to rotational distortion except for the anomalous states: 8,, 9,, 104 and 106, i.e. the relatively low T states. To account for this discrepancy we note that we have not included Coriolis forces which lead to energy changes that vary with the rotational state. Furthermore, this rotation-vibration interaction enters into the molecular hamiltonian via terms of the form II,&rr,. Rewrltigg this in a somewhat more transparent fashion, we have

where ji can be approximated as a diagonal, rigid tensor for the sake of simplicity. Here Qi and Pi refer to the vibrational modes. Since the bending mode involves the greatest deviation, we can take Qr to be parallel to the hydrogen-hydrogen separation. Taking Pi as one of the OH stretch modes, it immediately follows that the component of J that is parallel to the C-rotation axis is of greatest importance. However, J is parallel to this axis only for the low T states, and therefore it is these states that we describe most poorly with respect to the Coriolis interaction. In table VIII we have compared theory (correlated bender) and experiment for rotation-vibration levels that lie beyond J = 10. An examination of this table reveals that the model of a correlated bender accounts for these states within 1- 1.5% accuracy. In view of the fact that we have used only seven pieces of data, the demonstrated accuracy is quite satisfying. Note

Table VII Effects of centrifugal distortion

66 75 77 84 88 95 91 99 104 lo6 lo8 1010

CentrifugaI distortion (cm-‘)

Rigid bender (cm-l)

Amount

30.97 30.09 55.60 29.98 91.40 53.13 89.19 140.82 54.66 88.45 137.67 205.86

24.41 21.03 43.90 18.09 12.96 34.19 65.30 112.66 30.39 58.45 102.31 165.55

18.82 69.89 78.90 66.03 79.82 63.63 73.21 80.00 55.60 66.08 74.32 80.42

(%)

Correlated bender (cm-‘)

28.73 25.23 51.04 22.19 83.12 40.12 75.31 126.77 36.66 68.10 116.69 183.71

Amount (I)

92.76 83.85 91.80 14.02 90.94 75.79 84.40 90.02 61.07 77.00 84.16 a9.24

D. RogovinlA correlated bender approach to non-linear molecules Table VIII Comparison of high-energy rotation-viiration in the (000) state

experiment levels (in cm-l)

J*

Observed

Correlated bender

Dev. (%)

110

2143.01 2522.46 2938.36 3160.97 2437.62 2612.94 2813.61 2927.38 3128.25 3348.20 3584.00 2880.94 3101.57 3266.36 3083.92

2161.74 2549.74 2973.07 3216.60 2460.86 2640.63 2848.67 2963.20 3171.36 3398.26 3640.19 2900.34 3133.81 3309.54 3111.93

0.87 1.08 1.18 1.76 0.95 1.14 1.25 1.22 1.38 1.50 1.57 0.67 1.04 1.32 0.91

115 119 1111 120 122 1% 130

133 135 137 14-6 14-z 140 15-a

that many of these rotation-vibration states lie above 3000 cm-l which is about equal to the (020) state. We note that the large discrepancy of these energies in our two models of Hz0 cannot be accounted for in the differences of the diagonal rotation constants as they are nearly the same. Instead, one must account for this discrepancy via the off-diagonal elements of the rotation constants (see tables IV) which differ considerably in the two models. We note that these matrix elements govern the coupling between the various vibrational states arising from molecular rotation. Furthermore, these quantities cannot be directly measured and yield valuable information on the vibrational wave functions as they vary sensitively with the hydrogen-hydrogen motion. For example, for (01411)one has

@IAll)= +4ti&l

1_

(!,2)2 J/1(4)9

(2.19)

where $o[$~r] is the ground [(OlO)] state vibrational wave function and C is a constant that does not concern us here. Note, that for (Ok311)one has (01811)=

CJWo(4) (&

J/l(S).

121

(2.20)

Finally, in table IX we have compared theory and

[ 181 for the (010) state. In general, the correlated bender is still more accurate than the rigid bender. The overall accuracy for the correlated bender is on the order of l-1.5%.

4. The role of the nuclear masses In this section we examine the significance of the nuclear masses in our correlated bender model of water. This is best done by investigating the rotationvibration spectra and structure of the symmetric isotopic species D20 and T20. Specifically, we present calculations of the A, B and C rotation matrices as well as the rotation-vibration energy levels. By comparing these results with Hz0 a detailed picture of the role of the nuclear masses in the motion of the two hydrogen atoms can be obtained. Before presenting these results, we first note that our effective potential V(q) depends on the nuclear mass in two ways: (1) through the quantity U, and (2) through the correlated bender coordinate 4. The dependence of the P values on the nuclear masses is obvious and need not be discussed in detail here, although we will present below an order of magnitude estimate of the mass-dependence. On the other hand, the dependence of 4 on the nuclear mass is interesting and characteristic of the technique we are using to model water, and therefore must be discussed. We begin by noting that for any model of water, 4 corresponds to a one-dimensional path through the potential energy surface of the molecule. This path may be either linear or curved. For the rigid bender, r.~ is merely the bending coordinate itself and can be identified with the bending mode axis. On the other hand, in the correlated bender model of water, 4 represents a curved path. The degree to which this coordinate departs from a straight line parallel to the bending mode axis depends on the correlation between the bending mode and the OH stretch modes. To estimate the contribution of the first effect we again use the rigid bender model as a test. Specifically, we use the nuclear potential energy V,(q) generated in section 2 for Hz0 and calculate the first transition frequency of DzO. Since the coordinate Q is not an effective coordinate for this model the nuclear masses enter into V only through the reciprocal moments of inertia. We find that there is an additional deviation in the first transition frequency of D20 of -5 cm-1

D. Rogovin/A

122 Table IX Comparison of rotation-vibration

J7 11

20 22 30 33 40 42 44 50 53 55 60 62

64 66 70 73 75 ‘I 80 ‘34 88 90 95 97 99 100 104 lo6 lo8

correlated bender approach to non-linear molecules

energy levels (in cm-l)

in the (0, 1, 0) state

Observed

Rigid rotor

Rigid bender

Dev. (%)

Correlated bender

Dev. (%)

45.89 99.03 149.05 219.28 313.01 328.45 411.53 535.01 531.85 657.08 811.65 687.97 804.68 959.39 1139.65 1975.07 1129.71 1310.84 1515.43 1177.16 1506.69 1936.46 1545.06 1932.18 2157.99 2399.80 1793.08 2176.36 2403.21 2646.41

46.31 100.10 151.18 222.37 320.24 332.98 419.60 552.70 541.77 676.05 848.69 700.69 825.19 995.91 1206.60 998.42 1169.07 1387.80 1627.97 1205.13 1576.31 2112.27 1592.23 2046.59 2333.41 2659.52 1847.23 2293.98 2579.85 2695.82

46.17 98.83 150.79 221.01 317.43 328.44 415.36 542.48 535.80 664.82 822.69 690.00 812.11 970.16 1155.45 982.79 1142.30 1328.51 1538.68 1183.60 1526.13 1970.63 1559.82 1961.29 2196.16 2449.5 1 1807.81 2209.15 2446.04 2702.07

0.61 0.20 1.16 0.79 1.41 0.00 0.93 1.26 0.74 1.18 1.36 0.30 0.92 1.23 1.39 0.79 1.11 1.35 1.53 0.50 1.29 1.76 0.96 1.51 1.77 2.07 0.82 1.51 1.78 2.10

46.41 100.26 150.77 222.23 316.52 332.85 416.70 539.71 539.48 664.91 816.83 687.48 814.82 967.77 1144.59 989.82 1143.95 1321.98 1520.38 1196.23 1524.39 1941.97 1570.72 1954.05 2173.90 2406.75 1825.12 2207.98 2430.62 2667.14

1.13 1.24 1.15 1.35 1.12 1.34 1.26 0.89 0.68 1.19 0.64 0.07 1.26 0.98 0.43 1.51 1.26 0.85 0.33 1.62 1.17 0.28 1.66 1.13 0.74 0.29 1.79 1.45 1.14 0.78

so that the mass-dependence of U amounts to 0.4% of the transition energy. Furthermore, applying the correlated bender potential energy function V2(q), generated in section 2, to D20 we find a deviation of -11 cm-l from the correct transition frequency. Thus, taking into account U, the mass-dependence of the 4 coordinate amounts to about 0.5% of the transition frequency. Our procedure then, is to re-determine the potential energy function for both D20 and T20 so as to fit the first bending mode transition frequency. The results are displayed in table X. (The results for Hz0 are included for convenience.) We note that the first two terms, i.e. a2 and u4 which dominate the structure of the molecule, vary by 1% from H20 to D20. The a6 and aa terms display a greater variation; however, they

represent less than 0.1% of the energy associated with the a2 and a4 terms. Finally, all the odd parameters were set equal to zero and the ~2 transition was fitted exactly. Next we examine the various rotation matrices. As the rotation-vibration interaction is manifested in the off-diagonal components of the rotation matrices, Table X Potential energy constants display (cm-‘) Parameter

H2O

a2 a4 a6 a8

45 -3c) -23 1

D2O 510 833 665 820

46 -32 -20 -26

T2O 610 580 508 568

46 -32 -15 -22

590 147 002 736

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correlated bender approach to non-linear molecules

we confine our attention to these terms. Specifically we examine the VENOUS ratios AiilAii, BiilBii, and Cg/Cii, where Aij f I(ilA~>l,

which’are displayed in table XI. An examination of table XI reveals that there is a systematic trend, in particular, the corresponding offdiagonal matrix elements all decrease by about 80% when the two nuclear masses are increased by one unit each, e.g.

123

on the order of 0.6%, which is quite satisfactory in view of the fact that the only experimental data used in constructing this model of D20 was the first transition frequency (i.e. ~2) and the A and B ground state rotation constants [16].

5. Discussion

In this paper we have presented a model of water that inherently incorporates correlations between the bond angle and bond length vibrational modes. In particular, a detailed comparison between the corre(A,,M,&ao = 0~8(A,-,,/A,,),a, = 0.64(A01/AOO)Hz0. lated bender and the rigid bender demonstrates that these correlations constitute an important component Finally, in table XII we compare theory and experiof the rotation-vibration structure that arises from ment for ten rotation-vibration energy levels in the the non-bonded proton-proton motion. Specifically: ground and (010) excited vibrational states. In general, (1) The correlated bender yielded a better tit to agreement with the experimentally derived values is the vibrational spacings. Furthermore, the predicted ground state C rotation constant was closer to experiTable XI ment. Ratios of rotation matrix elements (2) The static bond angle and the equilibrium bond angle were closer to experiment in the correlated Parameter Hz0 D2O T20 bender model. Aolha 0.22 0.19 0.17 (3) Experiment demonstrates that the B rotation A oz/A oo 0.09 0.06 0.05 constant for the (010) state is greater than that of the AIZIAII 0.35 0.28 0.25 ground state. A model based on bond angle fluctuations only will always involve a decrease in B if A Bot&o 0.12 0.10 0.09 kz.&o 0.02 0.01 0.01 increases relative to the ground state. On the other WBII 0.18 0.15 0.13 hand, in the correlated bender model B increases as well as A. This feature of water is an excellent COlh 0.01 0.01 0.01 example of observable effects that arise from correco2lGcl -0 -0 -0 lations between the stretch and bending modes. GZlCll 0.02 0.02 0.02 (4) The A rotation constant in the correlated bender for the (010) state is significantly better than Table XII the rigid bender value. Rotation-vibration energy level structure in D20 (cm-‘) (5) The rotation-vibration energy level structure is significantly more accurate in the correlated bender Observed Theory Dev. (%) ” Jr model than one based on bond angle fluctuations 0 100 908.19 911.83 +0.40 alone. 0 102 1002.85 1007.65 +0.47 Thus, it is quite clear that the correlated bender 0 104 1114.85 1 i20.84 +0.53 model presents a much more accurate picture of the 1 104 1166.85 1177.64 +0.92 rotation-vibration structure and dynamics of water 0 lo6 1241.93 1249.16 +0.57 than a model based on fluctuations of the bond angle 1 lo6 1307.67 1317.18 +0.80 0 108 1382.69 1390.97 +0.60 alone. 1 lo8 1464.2 1470.2 +0.41 Next, we briefly compare our approach to water to 0 1010 1535.84 1545.14 +0.60 previous work. Darling and Dennison [l] presented 1 1010 1636.79 1635.11 -0.10 calculations of the molecular geometry and vibrational

124

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correlated bender approach to non-linear molecules

structure of H20 in 1940. Their work was based on a harmonic oscillator which was slightly perturbed by cubic terms and their results were in good agreement with experiment. No material was presented on the rotational-vibrational line spectra. Redlich [3] presented calculations on the anharmonic structure of the water molecule’s vibrational motion but did not investigate the rotational properties of H,O. More recently, Gordy et al. [ 121 and DeLucia et al. [lo] investigated the detailed rotational structure and molecular geometry of the ground state using a phenomenological rotational hamiltonian so chosen to fit the observed microwave spectral data. Their work represents an extremely important contribution to our knowledge of this molecule and much of their data was used as either a check on the accuracy of our model or for fitting purposes (specifically the ground state A and B rotation constants). Finally, Bunker et al. [4-61 have presented theoretical calculations of Hz0 in which the bending mode was treated as a kind of rotation. For single mode work based on fluctuations of the bond angle alone, their results are of the same order of accuracy as that of the rigid bender, although details of the high energy rotational states were not presented. In ref. 6 a semi-rigid bender model of water was presented Involving about 16 numerical constants. The results of this model are of the same order of accuracy as ours. Specifically, the rotational structure used in ref. 6 is somewhat more accurate than ours but the molecular geometry displays a somewhat larger discrepancy. In addition, this model employs many more parameters. We note that this approach differs considerably from ours as the stretch modes are directly incorporated in the potential. Our model is based on the notion of choosing a “classical trajectory”, which is so constructed as to include correlations of the stretch and bend modes. A comparison between the two approaches is interesting as it demonstrates that by including these correlations one can treat the ground and (010) states almost as well as a multi-mode approach. For example, in ref. 6 97% of the rotation-vibration interactions for the 1O1ostate is accounted for as compared to 9% in the

correlated bender. In summary, our model involves fewer parameters (only four) and exhibits in a clear manner the underlying physics that is responsible for the unique structure of water.

References [l] B. T. Darling and D. M. Dentin, Phys. Rev. 57 (1940) 128. [2] C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (McGraw-Hill, New York, 1955). [ 310. Redhch, J. Chem. Phys. 9 (1941) 298. [4] J. T. Hougen, P. R. Bunker and J. W. C. Johns, J. Mol. Spectrosc. 34 (1970) 136. [5] P. R. Bunker and J. M. R. Stone, J. Mol. Spectrosc. 41 (1972) 310. [6] A. R. Hoy and P. R. Bunker, J. Mol. Spectrosc. 52, (1974) 439. [7] D. Rogovin, M. Sargent, III and H. Tigelaar, Chem. Phys. Lett. 31(1975) 147. [8] F. DeLucia, P. Helminger, R. Cook and W. Gordy, Phys. Rev. A5 (1972) 487. [9] F. DeLucia, P. Helminger, W. Gordy, H. Morgan and P. Staats, Phys. Rev. A9 (1973) 2785. [lo] R. Cook, F. DeLucia and P. Helminger, J. Mol. Spectrosc. 53 (1974) 62. [ll] J. M. Flaud and C. Camy-Peyret, J. Mol. Spectrosc. 51 (1974) 142. [12] J. K. G. Watson, Mol. Phys. 15 (1968) 479. [13] J. K. G. Watson, MoL Phys. 19 (1970) 465. [14] The fact that the hamiltonians (2.2) and (2.5) are different is of no practical consequence as the probability that the molecule resides within a few degrees of the linear configuration is less than 0.1%. [ 151 M. Born and J. R Oppenheimer, Ann. Phys. 4 (1927) 457. [ 161 The energy level spacings were taken from W. S. Benedict, H. S. Classen and J. H. Shaw, J. Res. Nat. Bur. Stand. 49 (1954) 91. [ 171 One might argue that if one includes the seventh and eighth order terms to the potential V(9) we should include at least two more data points. We note, however, that for the low lying states that are of interest to us, terms of order (4 - qo)’ and (4 - Q,)’ play no significant role. ] 181 Taken from R. T. HalJ and J. M. Dowling, J. Chem. Phys. 47 (1967) 2454. [ 191 Data taken from L. A. Pugh and K. Narhari Rao, J. Mol. Spectrosc. 47 (1973) 403.