UDC 539.192 + 539.194

PHYSICS

M. R. ALIEV, V. T. ALEKSANYAN

ON THE CHOICE OF \(S\)-FUNCTIONS

IN THE METHOD OF CONTACT TRANSFORMATIONS

(Presented by Academician Ya. K. Syrkin, May 17, 1966)

1. The method of contact transformations (c.t.) was first applied by Van Vleck to the calculation of the multiplet structure of the levels of diatomic molecules \((^1)\), and subsequently by Jordahl in calculations of the paramagnetic susceptibility of a number of salts \((^2)\). Later this method was used mainly for the analysis of vibrational-rotational spectra of polyatomic molecules \((^3)\). In the works of Herman and Shaffer \((^4)\) and Amat, Nielsen, and Goldsmith \((^5,{}^6)\), \(S\)-functions of the first and second c.t. were found for diagonalizing the matrix of the vibrational-rotational energy of a polyatomic molecule. In the literature, \(S\)-functions are unknown for the subsequent transformations of the energy operator. Nor are there indications of general methods for finding them.

In the present work a general formula is derived for the c.t. of the operator of vibrational-rotational energy, and a method is proposed for finding \(S\)-functions for diagonalizing operators containing terms of the type \((p^m q^n + q^n p^m)\) with arbitrary \(m\) and \(n\).

2. Let a Hermitian operator \(H\) be given in the form

\[ H=\sum_{n=0}^{\infty}\lambda^n H_n, \]

where \(\lambda\) is a small parameter, and the orthonormal eigenbasis \(H_0(\Psi_0)\) and commutators of the type

\[ [H_m,H_n],\ [H_m,[H_n,H_k]],\ldots,\ m,n,k=0,\ldots,\infty \tag{1} \]

are known.

Owing to the Hermiticity of the operator \(H\), there exists some unitary transformation \(T\) that diagonalizes the matrix of the operator \(H\) given in the representation \(\Psi_0\), i.e.

\[ \Psi=\Psi_0 T,\qquad \Psi^{+}=T^{+}\Psi_0^{+},\qquad E=\Psi_0 THT^{+}\Psi_0^{+}. \tag{2} \]

Thus, the problem of determining the elements of the diagonal matrix \(E\) of the operator \(H\) reduces to transforming it by means of \(T\) into the operator \(THT^{+}\), diagonal in the representation \(\Psi_0\).

In the c.t. method \(T\) is specified in the form of the product

\[ T=T_{\infty}\cdots T_k T_{k-1}\cdots T_2 T_1, \tag{3} \]

and the individual \(T_k\) in the form

\[ T_k=\exp(i\lambda^k S_k),\qquad T_k^{+}=\exp(-i\lambda^k S_k), \tag{4} \]

which automatically ensures the orthonormality of the basis \(\Psi\). Expanding \(T_k\) in a series in powers of \(\lambda\), and restricting ourselves to the \(k\)-th term of (3), we obtain

\[ T_k=\sum_{n=0}^{\infty}(n!)^{-1}(i\lambda^k S_k)^n,\qquad T_k^{+}=\sum_{n=0}^{\infty}(n!)^{-1}(-i\lambda^k S_k)^n; \tag{5} \]

\[ T^{(k)}=\prod_k T_k=\prod_k \sum_{n=0}^{\infty}(n!)^{-1}(i\lambda^k S_k)^n; \tag{6} \]

\[ E^{(k)}=\Psi_0\left[\prod_k \sum_{n=0}^{\infty}(n!)^{-1}(i\lambda^k S_k)^n H \prod_k \sum_{n=0}^{\infty}(n!)^{-1}(-i\lambda^k S_k)^n\right]\Psi_0^+. \tag{7} \]

where the expression in square brackets in (7) is the \(k\)-fold transformed operator \((H^{(k)})\). The functions \(S_k\) must be chosen so that the operator \(H^{(k)}\) commutes with \(H_0\). Then the operators \(H^{(k)}\) and \(H_0\) have a common orthonormal basis \(\Psi_0\), and \(E^{(k)}\) is a diagonal matrix. Assigning to \(k\) the values \(1, 2, \ldots, n, \ldots\), one can successively obtain the corresponding \(E^{(n)}\), diagonal to within terms of the \((n+1)\)-st order of smallness.

For \(k=1\), for \(H^{(1)}\) we obtain the expression

\[ \begin{aligned} H^{(1)} &=\sum_{n=0}^{\infty}\lambda^n H_n^{(1)} =\sum_{n=0}^{\infty}\lambda^n H_n +\sum_{l=1}^{\infty}(l!)^{-1}(i\lambda)^l \left[S_1,\underbrace{[\ldots [}_{l\ \text{times}}S_1, \sum_{m=0}^{\infty}\lambda^m H_m]\ldots]\right] \\ &=\sum_{n=0}^{\infty}\lambda^n H_n +\sum_{l=1}^{\infty}\sum_{m=0}^{\infty}(l!)^{-1} i^l \lambda^{l+m} K_{lm}^{(1)}, \end{aligned} \tag{8} \]

where the commutators \(K_{lm}^{(1)}=[S_1,\underbrace{[\ldots [}_{l\ \text{times}}S_1,H_m]\ldots]]\) are expanded according to the usual rules.

For arbitrary \(k\),

\[ H^{(k)}=\sum_{n=0}^{\infty}\lambda^n H_n^{(k-1)} +\sum_{l=1}^{\infty}\sum_{m=0}^{\infty}(l!)^{-1} i^l \lambda^{lk+m} K_{lm}^{(k)}, \tag{9} \]

where \(K_{lm}^{(k)}=[S_k,[\ldots [S_k,H_m^{(k-1)}]\ldots]]\). Expressions for the individual \(H_n^{(k)}\) can be obtained by equating terms with identical powers \(\lambda^n\) in the right- and left-hand sides of (9). All \(S\)-functions, up to \(S_k\), are chosen so that the operator \(H^{(k)}\) is diagonal in the basis \(\Psi_0\). The terms \(H_n^{(k)}\) with \(n>k\) are nondiagonal in this basis \(\Psi_0\), but the nondiagonal elements \(H_n^{(k)}\) (with \(n>k\)) give a correction to the eigenvalues of \(H_0\) only in the \((n+1)\)-st approximation. Therefore, to compute corrections to the eigenvalues of \(H_0\) in the \(n\)-th approximation it is sufficient to make \(n-1\) transformations, whereas to compute corrections to the eigenvectors in the same (\(n\)-th) approximation it is necessary to make \(n\) transformations of \(H\).

1. Suppose all \(S\)-functions up to \(S_{k-1}\) have been found and all operators \(H_n^{(k-1)}\) with \(n\le k\) have been constructed. It follows from equation (9) that the function \(S_k\) must satisfy the condition

\[ H_k^{(k)}=H_k^{(k-1)}+iK_{10}^{(k)}, \tag{10} \]

where \(H_k^{(k)}\) in the basis \(\Psi_0\) is a diagonal matrix.

Finding a function \(S_k\) satisfying condition (10) is possible if the commutation laws are known for the operators of which \(H_k^{-1}\) is composed, with \(H_0\). For the vibrational-rotational energy operator of a diatomic molecule [7], the problem is simplified by the fact that \(H_k^{(k-1)}\) is expressed through combinations of operators of the dimensionless coordinate and momentum of the type \((p^m q^n+q^n p^m)\). Using the relation \([p,q]=-i\), one can obtain commutation relations for any combination \((p^m q^n+q^n p^m)\) with \(H_0\).*

* The commutation relations for \((p^m q^n+q^n p^m)\) used below in the calculations are not given for lack of space.

Table 1

To choose the function \(S_k\) that diagonalizes \(H_k^{(k-1)}\), we shall use the following device.

Let us specify \(S^k\) in the form of a linear combination of the operators \((p^l q^j+q^j p^l)\)

\[ S_k=\sum_{lj} y_{lj}(p^l q^j+q^j p^l), \qquad l+j \leqslant m+n, \tag{11} \]

and \(H_k^{(k)}\) in the form of the series

\[ H_k^{(k)}=\sum_i x_i(p^2+q^2)^i, \qquad i \leqslant (m+n)/2, \tag{12} \]

where the operator \((p^2+q^2)\), and hence all its powers, are diagonal in the vibrational quantum number \(v\). Substituting (11) and (12) into (10) and equating the coefficients of identical combinations \((p^m q^n+q^n p^m)\) on the right- and left-hand sides of (10), one can find all \(y_{lj}\) and \(x_i\), and consequently the function \(S_k\) and, at the same time, the diagonal correction \(H_k^{(k)}\).

If \(H_k^{(k-1)}\) contains many terms, they can be diagonalized separately: the complete function \(S_k\) will be equal to the sum of the \((S_k)_i\) that diagonalize the separate terms \((H_k^{(k-1)})_i\) in \(H_k^{(k-1)}\), and the diagonal correction \(H_k^{(k)}\) will be the sum of the diagonal corrections \((H_k^{(k)})_i\).

As an example, let us find the function \((S_k)_i\) for diagonalizing \((H_k^{(k-1)})_i=c_i q^6\), where \(c_i\) is a constant coefficient. We represent \((S_k)_i\) and \((H_k^{(k)})_i\) in the following form

\[ (S_k)_i=y_{51}(p^5q+qp^5)+y_{15}(pq^5+q^5p)+y_{33}(p^3q^3+q^3p^3)+y_{11}(pq+qp), \]

\[ (H_k^{(k)})_i=x_1(p^2+q^2)+x_3(p^2+q^2)^3. \tag{13} \]

The remaining coefficients \(y_{lj}\) and \(x_i\) are equal to zero.*

Substituting (13) and \((H_k^{(k-1)})_i=c_iq^6\) into (10) and taking into account that

\[ [(p^5q+qp^5),(p^2+q^2)]=4ip^6-10i(p^4q^2+q^2p^4)-60ip^2, \]

\[ [(pq^5+q^5p),(p^2+q^2)]=-4iq^6+10i(p^2q^4+q^4p^2)+60iq^2, \]

\[ [(p^3q^3+q^3p^3),(p^2+q^2)]=6i(p^4q^2+q^2p^4)-6i(p^2q^4+q^4p^2)+18i(p^2-q^2), \]

\[ x_3(p^2+q^2)^3=x_3(p^6+q^6)+\frac{3}{2}x_3(p^4q^2+q^2p^4+p^2q^4+q^4p^2)+ \]

\[ +4x_3(p^2+q^2), \]

we obtain the following values for the coefficients \(y_{lj}\) and \(x_i\):

\[ y_{51}=-\frac{5}{64}\frac{2c_i}{\omega}; \qquad y_{15}=-\frac{11}{64}\frac{2c_i}{\omega}; \qquad y_{33}=-\frac{5}{24}\frac{2c_i}{\omega}; \]

\[ y_{11}=-\frac{15}{16}\frac{2c_i}{\omega}; \qquad x_3=\frac{5}{16}c_i; \qquad x_1=\frac{25}{16}c_i. \]

In this way the \(S\)-functions given in the table have been found for diagonalizing the operators \((p^m q^n+q^n p^m)\) for values \(m+n\leqslant 6\). The first column of the table contains the various \((p^m q^n+q^n p^m)\) entering as separate terms in \(H_k^{(k-1)}\), the second contains the function \((S_k)_i\), and the third contains the diagonal correction \((H_k^{(k)})_i\). We note that the diagonal correction from \((p^m q^n+q^n p^m)\) is equal to zero if at least one of the exponents \((m\) or \(n)\) is odd. For even \(m\) and \(n\) in (12), only the terms with \(i=(m+n)/2,\ (m+n)/2-2\), etc., are nonzero.

The authors express their deep gratitude to L. S. Mayants for a useful discussion of the results of the work.

Institute of Organoelement Compounds
Academy of Sciences of the USSR

Received
10 V 1966

CITED LITERATURE

1. J. H. Van Vleck, Phys. Rev., 33, 467 (1929).
2. J. M. Johrdahl, Phys. Rev., 48, 87 (1934).
3. H. H. Nielsen, Rev. Mod. Phys., 23, 90 (1951).
4. R. C. Herman, W. H. Shaffer, J. Chem. Phys., 16, 453 (1948).
5. G. Amat, M. Goldsmith, H. Nielsen, J. Chem. Phys., 27, 838 (1957).
6. G. Amat, H. Nielsen, J. Chem. Phys., 27, 845 (1957).
7. H. Hanson et al., J. Chem. Phys., 27, 40 (1957).

* The coefficients \(y_{lj}\) that are different from zero are chosen with the aid of the table of commutators \([(p^m q^n+q^n p^m),(p^2+q^2)]\).