UDC 539.194:621.371.166.2

PHYSICS

A. A. VIKTOROVA, S. A. ZHEVAKIN

THE WATER VAPOR DIMER AND ITS SPECTRUM

(Presented by Academician M. A. Leontovich, February 8, 1966)

To calculate the rotational spectrum of the water-vapor dimer we use a linear model of the dimer, the structure of which was considered in ((¹), p. 415). This is a compound with internal rotation: one water molecule undergoes hindered rotation relative to the other molecule about the line of the hydrogen bond O ... O₂ in the potential field \(V(\alpha)=\frac{1}{2}V_0(1-\cos\alpha)\), where \(\alpha\) is the relative angle of rotation of the molecules about the bond line; \(V_0\) is the height of the potential barrier, equal to 455 cm\(^{-1}\) (1.3 kcal/mole).

Burkhard and Irwin (²), using the method of internal axes (³–⁵), considered the general case of two asymmetric tops undergoing hindered rotation relative to one another (the water-vapor dimer is precisely such a compound). The Hamiltonian can be represented as the sum of three terms \(\hat H=\hat H_1+\hat H_2+\hat H_3\), where \(\hat H_1\) is the Hamiltonian of an effective symmetric top, which describes the rotation of both groups forming the dimer, or the “external” rotation; the operator \(\hat H_2\) describes the “internal,” or hindered, rotation in the potential field \(V(\alpha)\); the operator \(\hat H_3\), due to the asymmetry of the compound, describes the interaction of the “external” and “internal” rotations.

We calculate the matrix elements of the Hamiltonian with the aid of the basis eigenwave functions of the operator \(\hat H_0=\hat H_1+\hat H_2\), equal to

\[ U_{JKMv}=\frac{1}{2\pi} e^{iK\varphi} e^{iM\psi}\Theta_{JKM}(\theta) P_{vK}(\alpha)\exp\left[-\,i\frac{C_1}{(C_1+C_2)}K\alpha\right]. \tag{1} \]

Here \(\theta,\varphi,\psi\) are the Euler angles; moreover

\[ \varphi=\frac{C_1}{(C_1+C_2)}\varphi_1+\frac{C_2}{(C_1+C_2)}\varphi_2, \]

where \(\varphi_1\) and \(\varphi_2\) are the angles of rotation, respectively, of the first and second water molecules about the internal-rotation axis; \(C_1\) and \(C_2\) are the moments of inertia about this axis for the first and second water molecules. The factor \(e^{iK\varphi}e^{iM\psi}\Theta_{JKM}(\theta)\) in formula (1) is the usual wave function of a symmetric top (\(J\) is the quantum number of the total angular momentum; \(K\) and \(M\) are the quantum numbers of the projections of the angular momentum, respectively, on the internal-rotation axis and on an axis fixed in space). The factor \(F_{vK}(\alpha)=P_{vK}(\alpha)\exp\left[-\,i\frac{C_1}{(C_1+C_2)}K\alpha\right]\) is the wave function for internal rotation, satisfying the equation for a hindered rotor (⁶); \(v\) is the torsional quantum number.

The matrix elements of the interaction operator \(\hat H_3\) in this representation are small and we shall not take them into account. The eigenvalues of the operator \(\hat H_0\) are equal to the sum of two terms. The first term, describing the “external” rotation, is the rotational energy of an effective symmetric top and is given by the expression

\[ E_{JK}=\frac{\hbar^2}{2}\left\{\frac{1}{4N'}\left[C_1C_2(A_1+A_2+B_1+B_2)-C_1E_2^2-C_2E_1^2\right](J^2+J-K^2)+\frac{K^2}{C}\right\}, \]

where \(A_1, A_2, B_1, B_2, C_1, C_2, C, E_1, E_2\) are moments and products of inertia, defined in (2); \(N'\) is a function of these quantities (see ibid.); \(h\) is Planck’s constant.

The data on the dimer configuration needed for estimating these constants were taken from \((^{1a})\). We also note that, in calculating \(E_{JK}\), the displacement of the dimer center of inertia relative to the axis of internal rotation was neglected because of its smallness. As a result, taking into account the parameters of the \(\mathrm{H_2O}\) dimer, the energy \(E_{JK}\) can be written in the form

\[ E_{JK}=0.23J(J+1)+7.07K^2\ \left(\mathrm{cm}^{-1}\right). \]

The second term, \(E_{vK}\), characterizing the energy of internal rotation, is equal to \((^6)\)

\[ E_{vK}=\frac{4V_0}{s}\sum_{l=0}^{\infty} a_l \cos\left[2\pi\frac{C_1}{(C_1+C_2)}Kl\right]. \]

Here the coefficients \(a_l\) depend on the torsional quantum number \(v\) and the reduced barrier height \(s\) (see \((^{6,7})\)); for the dimer \(s=60\). Using the tables of Ref. \((^8)\), the values of \(E_{vK}\) were found for \(s=60\) and \(v=0,1,2\):

\[ E_{0K}=58.2\ \left(\mathrm{cm}^{-1}\right),\qquad E_{1K}=170.3\ \left(\mathrm{cm}^{-1}\right), \]

\[ E_{2K}=\left\{272.9-0.1477\cos\left[2\pi\frac{C_1}{(C_1+C_2)}K\right]\right\}\ \left(\mathrm{cm}^{-1}\right). \tag{2} \]

As is evident, the Fourier series in the expression for \(E_{vK}\) converges very rapidly, and \(E_{vK}\) depends only weakly on \(K\). This is true when the barrier to internal rotation is sufficiently high. The barrier belongs to the category of “high” barriers if \(s\geq 20\) (or \(V_0\geq 0.7\ \mathrm{kcal/mol}\) \((^6)\)). In this case (which also applies to the dimer), internal rotation may be considered in the harmonic-oscillator approximation for energies \(E_{vK}\) not exceeding the barrier height. For the \(\mathrm{H_2O}\) dimer, the number of molecules with energy \(E_{vK}\geq V_0\) constitutes no more than 5% of the total number of dimers.

In the harmonic-oscillator approximation, the energy of internal rotation reduces to the energy of torsional vibrations \(E_v=200(v+1/2)\ \left(\mathrm{cm}^{-1}\right)\). It is seen that this approximation agrees fairly well with the exact solution for \(v\leq 2\) (see the values (2) given above). It is difficult to make an equally definite statement about the energies with quantum number \(v>2\), owing to the absence of tables for \(a_l\) at \(v>2\). In what follows, when considering internal rotation, we shall approximate the exact wave functions \(F_{vK}(\alpha)\) by the harmonic-oscillator wave functions \(\mathcal{H}_v(\alpha)\), and take the values (2) as the energies.

The absorption coefficient \(\gamma\) can be calculated from formula \((^7)\)

\[ \gamma\left[\frac{\partial\delta}{KM}\right] = 10^6\log_{10}e\, \frac{32\pi^2N}{3hcG(T)\lambda^2} \sum_{ij}|\mu_{ij}|^2\, \frac{1}{\lambda_{ij}} \times \]

\[ \times \left|e^{-E_i/kT}-e^{-E_j/kT}\right| \frac{(\Delta\nu/c)_{ij}^{0}\sqrt{T/293}\,(P/760)} {\left[(1/\lambda_{ij})^2-(1/\lambda)^2\right]^2 + 4\left[(\Delta\nu/c)_{ij}^{0}\sqrt{T/293}\,(P/760)\right]^2(1/\lambda)^2}. \tag{3} \]

Here \(i\) and \(j\) denote the set of quantum numbers corresponding to nondegenerate rotational quantum states; the summation is carried out over all quantum transitions; \(\lambda\) is the wavelength of the electromagnetic radiation; \(\lambda_{ij}\) is the resonance wavelength of the dimer molecule for the transition from state \(i\) to state \(j\); \(E_i\) and \(E_j\) are the energies of the \(i\)- and \(j\)-states, respectively; \((\Delta\nu/c)_{ij}^{0}\) is the half-width of the spectral line in \(\mathrm{cm}^{-1}\) at absolute temperature \(T=293^\circ\) and pressure \(P=760\ \mathrm{mm\,Hg}\) for the transition \(i\to j\); the factor \(\sqrt{T/293}\) determines the temperature dependence of the half-widths of the spectral lines of the dimer under the assumption that the effective collision cross section of the dimer with surrounding molecules does not depend on their relative velocity; \(h, c\), and \(k\) are, respectively, the constants

Planck’s constant, the speed of light, and Boltzmann’s constant; \(G(T)\) is the statistical sum, equal to

\[ G(T)=\sum_{JKv} g_I(2J+1)\exp\left[-\frac{(E_{JK}+E_{vK})}{kT}\right], \]

where \(g_I\) is the nuclear statistical weight. For a random symmetric top (which the \(\mathrm{H_2O}\) dimer is), nuclear spin increases the statistical weight by the same factor for all rotational levels, as a result of which the factor \(g_I\) does not enter into the expression for the absorption coefficient. If inversion doubling is also not taken into account, then the statistical weight of levels with \(K=0\) will be equal to \((2J+1)\), and the statistical weight of levels with \(K\ne 0\) will be \(2(2J+1)\). In the expression for \(\gamma\) the form of the spectral line obtained from the solution of the kinetic equation \((^{9,10})\) was used. The quantity \(N\) in (2), \(N=10^{-6}\rho_d N_0/\eta\), is the number of water-vapor dimer molecules in \(1\ \mathrm{cm^3}\). Here \(N_0=6.025\cdot 10^{23}\) is Avogadro’s number; \(\eta=36\) is the molecular weight of the dimer; \(\rho_d\) is the concentration of dimers (in \(\mathrm{g/m^3}\)). In calculating \(\gamma\), the quantity \(\rho_d\) was computed for a hydrogen-bond energy equal to \(5.2\ \mathrm{kcal/mol}\) \((^{11})\), at an absolute air humidity \(\rho_{\mathrm{H_2O}}=7.5\ \mathrm{g/m^3}\), and at \(T=293^\circ\). As a result, from formulas \(((16),\) p. 424) we obtain \(\rho_d=0.022\ \mathrm{g/m^3}\), i.e., the number of dimers in the atmosphere is about \(0.1\%\) of the number of water-vapor monomers. The quantity \(|\mu_{ij}|^2\) in the expression for \(\gamma\) is the square of the modulus of the matrix element of the dipole moment for the transition \(i\to j\). The water-vapor dimer possesses a component of the electric dipole moment not only in the direction of the symmetry axis of the effective symmetric top (the axis of internal rotation), but also in the direction perpendicular to it, since the top is a random symmetric rotor and not due to the symmetry of the compound.

Because the basis wave function \(U_{JKMv}(\alpha)\) is equal to the product of functions describing, respectively, the symmetric top (a function of the Euler angles) and the hindered rotor \((F_{vK}(\alpha))\), the quantity \(|\mu_{ij}|^2\) can be written in the form

\[ |\mu_{ij}|^2=\frac{\beta_{ij}}{(2J+1)} \left\{\langle F_{v_iK_i}(\alpha)|\mu|F_{v_jK_j}(\alpha)\rangle\right\}^2, \]

where the quantum number \(J\) refers to the \(i\)-th state, \(\langle F_{v_iK_i}(\alpha)|\mu|F_{v_jK_j}(\alpha)\rangle\) is the matrix element of the modulus of the electric moment causing the transition \(i\to j\), obtained with the aid of the wave functions of internal rotation \(F_{vK}(\alpha)\); \(\beta_{ij}\) is the oscillator strength of the quantum transition \(J_i,\ K_i\to J_i,\ K_J\) for a rigid symmetric top (the values of \(\beta_{ij}\) for \(J,\ K<12\), obtained from rigorous formulas, were taken from tables \((^{12})\); for quantum numbers \(J,\ K\ge 12\), the quantities \(\beta_{ij}\) were calculated by us from tables \((^{12})\) by means of finite-difference calculation).

The selection rules for radiative transitions between the energy levels of a symmetric top with hindered internal rotation for the component of the electric dipole moment lying along the axis of internal rotation are: \(\Delta J=\pm 1;\ \Delta K=0;\ \Delta v=0\). In this case the quantity \(\left\{\langle F_{v_iK_i}(\alpha)|\mu_{\parallel}|F_{v_iK_i}(\alpha)\rangle\right\}^2=|\mu_{\parallel}|^2=4.1774\cdot 10^{-36}\ (\mathrm{CGSE})^2\). Using the constants of the \(\mathrm{H_2O}\) dimer, one may say that these transitions give a series of lines \((\nu/c)_{ij}=0.46\,J\ (\mathrm{cm^{-1}})\), characteristic of symmetric rotors.

For the component of the dipole moment perpendicular to the axis of internal rotation, the selection rules are: \(\Delta J=\pm 1,\ 0;\ \Delta K=\pm 1\); the quantity \(\Delta v\) is indeterminate. However, for high potential barriers (more precisely, for levels \(E_{vK}<V_0\), when the harmonic-oscillator approximation is valid), internal rotation is close to torsional vibrations and \(\Delta v=\pm 1,\ 0\).

Consider the transition \(J, K, v \to J', K-1, v'\). Using the expressions for the energies \(E_{JK}\) and \(E_{vK}\), and taking into account that \(J \geq K\), the frequencies of such transitions can be written as follows (in \(\mathrm{cm}^{-1}\)):

\[ (\nu/c)_{J\to J-1}=\nu'/c+0.46J;\qquad J=K,\ K+1,\ K+2,\ldots \qquad (P\text{-branch}), \]

\[ (\nu/c)_{J\to J}=\nu'/c \qquad (Q\text{-branch}), \]

\[ (\nu/c)_{J\to J+1}=\nu'/c-0.46J;\qquad J=K+1,\ K+2,\ldots \qquad (R\text{-branch}), \]

where

\[ \nu'/c=7.07(2K-1)+(E_{vK}-E_{v'K-1})/hc . \]

Transitions for which \(K-1 \to K\) will have the same structure. Let us note that in the \(P\)- and \(R\)-branches the distance between the lines corresponding to one and the same value of \(v\) is constant and, for the \(\mathrm{H_2O}\) dimer, is equal to \(0.46\ \mathrm{cm}^{-1}\); the lines of the \(Q\)-branch (which are the most intense) for one and the same value of \(v\) are separated from one another by \(14.14\ \mathrm{cm}^{-1}\). Each of the \(Q\)-branch lines in fact consists of a large number of individual “sublines,” corresponding to different values of the quantum number \(J\) (which, for small \(K\), can take large values up to \(J=70\)).

In what follows we shall be interested in transitions with \(\Delta v=0\). They correspond to the purely rotational spectrum of the symmetric top. Neglecting the interaction of rotation with torsional vibrations (as well as with other vibrations), we obtain for transitions \(\Delta K=\pm1\) and \(\Delta v=0\) that the quantity

\[ \{\langle F_{viK_i}(\alpha)|\mu_{\perp}|F_{viK_j}(\alpha)\rangle\}^{2} \simeq \{\langle \mathcal{H}_{vi}(\alpha)|\mu_{\perp}|\mathcal{H}_{vi}(\alpha)\rangle\}^{2}. \]

For the value of this quantity averaged over torsional states with energies \(E_v<V_0\), we obtain \(6.5\cdot10^{-36}\ \mathrm{(CGSE)}^{2}\). In calculating the matrix elements of the dipole moment of the dimer, the dipole moment of the \(\mathrm{H_2O}\) monomer was taken to be \(1.8\cdot10^{-18}\ \mathrm{CGSE}\).

Transitions with \(\Delta v=\pm1\) will correspond to the rotational–vibrational spectrum of the dimer. They lie in the region \((\nu/c)_{ij}\sim100\ \mathrm{cm}^{-1}\) and require special consideration. Transitions with \(\Delta v>1\) lie in the region \((\nu/c)_{ij}>100\ \mathrm{cm}^{-1}\) and are less intense.

For energies \(E_{vK}>V_0\) (which is realized for approximately \(5\%\) of the \(\mathrm{H_2O}\) dimers), the internal rotation may be regarded as free. It can be shown that such molecules contribute to the absorption only for \((\nu/c)>90\ \mathrm{cm}^{-1}\).

Scientific Research Radiophysics Institute
at Gorky State University
named after N. I. Lobachevsky

Received
7 II 1966

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