UDC 539.01

PHYSICS

Academician S. V. VONSOVSKII, V. I. CHEREPANOV,
A. N. MEN’, A. E. NIKIFOROV

GROUP-THEORETICAL CLASSIFICATION OF THE STATES OF PAIRS AND COMPLEXES OF IMPURITY PARAMAGNETIC IONS IN CRYSTALS

In a previous article by the authors \((^{1})\), a group-theoretical method was proposed for determining the allowed terms of a pair of bound impurity ions in a crystal. In the present article this method is generalized to a symmetric complex of \(N\) impurity ions in an ionic crystal. In addition, some inaccuracies of the preceding article are eliminated.

If covalent effects are neglected, then the electrons of the filled shells of nonimpurity ions are immaterial for the classification of the states of the complex.* Therefore let us consider the Hamiltonian of a system of electrons belonging only to the unfilled shells of \(N\) impurity paramagnetic ions:

\[ H=\sum_{i=1}^{N} H_0(i)+\sum_i \sum_{j\ne i} \bar V_j(i)+V_{ee}+\sum_i V_{\mathrm{cr.env}}(i), \tag{1} \]

where \(H_0(i)\) is the Hamiltonian of the \(i\)-th free impurity ion; \(\bar V_j(i)\) is the energy of the \(i\)-th impurity ion in the field produced by the nucleus (and the filled electron shells) of the \(j\)-th impurity ion; \(V_{ee}\) is the energy of interaction of electrons of different impurity ions with one another; and \(V_{\mathrm{cr.env}}(i)\) is the energy of the \(i\)-th impurity ion in the crystal field produced by the matrix ions surrounding the impurity complex.

The first three terms in (1) have the symmetry \(G_C^m\) of the “impurity molecule” (with a “center” at some point \(C\)), while the last term has the symmetry \(G_C\) of the crystal field produced by the crystal with “holes” at the sites occupied by substitutional impurity ions (the symmetry of the environment of the complex). Therefore we shall define the symmetry group \(G_k\) of the complex in the crystal as the intersection of the groups:

\[ G_k=G_C^m \cap G_C \tag{2} \]

(for a pair \(D_C^m=D_{\infty h}\) or \(C_{\infty v}\)).

In the Hamiltonian (1) one can always single out a zeroth-approximation term** \(\sum_i [H_0(i)+V_{\mathrm{cr}}(i)]\), adding and subtracting the corresponding summands, where \(V_{\mathrm{cr}}(i)\) is the energy of the impurity ion in the crystal field produced by the principal ions of the crystal. This term has the symmetry \(G_1 \times G_2 \times \cdots \times G_N\) (\(G_i\) is the point symmetry group of the \(i\)-th ion in the absence of other impurity ions). We shall start from the specified states \(\alpha_i\Gamma_i\) of the impurity ions in the crystal field with symmetry \(G_i\) (including spin-orbit interaction). Here \(\Gamma_i\) is an irreducible representation of the group \(G_i\), and \(\alpha_i\) are additional quantum numbers.

* Consideration of covalent effects in our scheme is possible, for example, by including in the complex under consideration, besides impurity ions, also the ligands nearest to them and taking account of all ionic and covalent structures of such a complex.

** For classification of a state, the order of magnitude of the various terms of the Hamiltonian is not important.

Let us denote the wave function of ion \(A_i\) by \(\psi_{A_i t_i a_i \Gamma_i \mu_i}(q_i)\), where \(t_i\) is the ion type (taking into account its species and crystal environment); \(\mu_i\) is a “row” of the representation \(\Gamma_i\); \(q_i\) is the set of Cartesian and spin coordinates of all \(n_i\) electrons of ion \(A_i\).

To each element \(g \in G_k\) there corresponds a certain permutation of impurity ions, which can always be represented as a product of cyclic permutations
\(g \to P_g = P_g(A_1, A_2, \ldots, A_m)\ldots\)
(the dots at the end denote other cyclic permutations). Thus, for each \(g\) all impurity ions are naturally distributed into sets—cycles.

The antisymmetric wave functions of the states of the complex (in the zeroth approximation), belonging to the eigenvalue
\(\sum_i E^{(0)}(A_i t_i a_i \Gamma_i)\), can be written in the form

\[ \Psi_{A_1 t_1 a_1 \Gamma_1 \mu_1 \ldots A_m t_m a_m \Gamma_m \mu_m;\ldots} (q_1\ldots q_m;\ldots) = \sum_{\mathscr P}(-1)^{\mathscr P}\mathscr P\, \psi_{A_1 t_1 a_1 \Gamma_1 \mu_1}(q_1)\ldots \psi_{A_m t_m a_m \Gamma_m \mu_m}(q_m)\ldots , \tag{3} \]

where \(\mathscr P\) are the operators of all possible permutations of electron coordinates between impurity ions (it is clear that only equivalent ions can enter one and the same cycle, i.e., ions of the same type \(t_1=t_2=\cdots=t_m=t;\ldots\)). We note that the set of wave functions (3) can be divided into separate sets that differ from one another by all possible permutations of unequal states among equivalent ions (including those from different cycles). These permutations take into account the expression for the exchange of states between equivalent ions. Each set contains also \([\Gamma_1][\Gamma_2]\ldots[\Gamma_N]\) functions, differing from one another by the values \(\mu_1,\mu_2,\ldots,\mu_N\) (\([\Gamma_i]\) is the dimension of the representation \(\Gamma_i\)).

The wave functions of the ions are each defined in its own local coordinate system \(K_i\). Having chosen the local coordinate system \(K_1\) of ion \(A_1\) of the first cycle, one may choose the local coordinate systems of the remaining ions, for example, by the action of the operator \(\hat g\):
\(K_2=\hat g K_1,\ K_3=\hat g K_2,\ldots, K_m=\hat g K_{m-1}\)
(analogously for other cycles). We note that with such a choice
\(K_1=\hat g^{-m}\cdot \hat g\cdot K_m\), i.e., in the case when
\(\hat g^m\ne E\), we have \(K_1\ne \hat g K_m\). It is not difficult to show that, with our choice of local coordinate systems, the action of the element \(\hat g\) on the wave functions of the ions reduces to the following:

\[ \hat g\psi_{A_i t_i a_i \Gamma_i \mu_i}(q_i) = \psi_{A_{i+1} t_i a_i \Gamma_i \mu_i}(q_i) \qquad \text{for } i=1,2,\ldots,m-1, \]

\[ \hat g\psi_{A_m t_m a_m \Gamma_m \mu_m}(q_m) = \hat g^m \psi_{A_1 t a_m \Gamma_m \mu_m}(q_m) \qquad \text{for } i=m. \tag{4} \]

Let us now consider the action of the element \(\hat g\) on the functions (3) and, taking the set (3) as a basis of a representation of the group \(G_k\), derive formulas for determining the characters of this representation. Using (4), we find*

\[ \hat g\Psi_{A_1 t a_1 \Gamma_1 \mu_1 \ldots A_m t a_m \Gamma_m \mu_m;\ldots} (q_1\ldots q_m;\ldots) = \]

\[ = \sum_{\mathscr P}(-1)^{\mathscr P}\mathscr P \left[ \sum_{\mu_m'}D^{(\Gamma_m)}_{\mu_m'\mu_m}(\hat g^m) \hat P_g(A_1A_2\ldots A_m)\, \psi_{A_1 t a_1 \Gamma_1 \mu_1}(q_1)\ldots \right. \]

\[ \left. \ldots \psi_{A_m t a_m \Gamma_m \mu_m'}(q_m) \right]\ldots \]

* Here it is taken into account that the element \(g^m\) returns all ions of the cycle to their initial positions, i.e., this element must be present in each of the equivalent point groups \(G_1, G_2,\ldots,G_m\).

The antisymmetric wave function will not change if the coordinate permutation operator \(P_g(q_1q_2\ldots q_m)\) is applied to it and this function is simultaneously multiplied by \(\pi^n=(-1)^{wn}\), where \(n\) is the number of electrons in each of the ions of the cycle, and \(w\) is the parity of the cyclic permutation \(P_g\). Analogous operations must be performed for the other cycles. As a result, taking also into account that application of the operator \(P_g(A_1A_2\ldots A_m)\cdot P_g(q_1q_2\ldots q_m)\) in our case is equivalent to application of the operator \(P_g^{-1}(a_1\Gamma_1\mu_1, a_2\Gamma_2\mu_2,\ldots, a_m\Gamma_m\mu'_m)\), we obtain:

\[ \hat g\Psi_{A_1t\alpha_1\Gamma_1\mu_1\ldots A_mt\alpha_m\Gamma_m\mu_m;\ldots}(q_1\ldots q_m;\ldots) = [\pi^n\ldots]\left[\sum_{\mu'_m}D_{\mu'_m\mu_m}^{(\Gamma_m)}(g^m)\ldots\right]\times \]

\[ \times \left[P_g^{-1}(a_1\Gamma_1\mu_1,\ldots,a_m\Gamma_m\mu'_m)\ldots\right] \Psi_{A_1t\alpha_1\Gamma_1\mu_1\ldots A_mt\alpha_m\Gamma_m\mu'_m;\ldots} (q_1q_2\ldots q_m). \tag{5} \]

If, in at least one of the cycles, not all \(a_i\Gamma_i\mu_i\) are identical, then the original function will not enter the right-hand side of (5), since the states \(a_i\Gamma_i\) of the ions are cyclically permuted under the action of the operator \(P_g^{-1}\). Therefore the contribution to the character of those sets of functions from (3) to which unequal states \(a_i\Gamma_i\mu_i\) of the ions correspond, in at least one of the cycles, will be equal to zero. The remaining sets give the character

\[ X(g)=[\pi^n\ldots \bar\pi^{\bar n}] \sum_{\text{over sets}}^{\prime} X^{(\Gamma)}(g^m)\ldots X^{(\bar\Gamma)}(\bar g^{\bar m}). \tag{6} \]

(the quantities pertaining to the last cycle are marked with a bar above), where the summation extends over all sets, and the prime denotes the condition
\[ a_1\Gamma_1=a_2\Gamma_2=\ldots=a_m\Gamma_m=a\Gamma;\ \ldots;\ \bar a_1\bar\Gamma_1=\bar a_2\bar\Gamma_2=\ldots=\bar a_{\bar m}\bar\Gamma_{\bar m}=\bar a\bar\Gamma. \]
(If there are no sets satisfying this condition, the character is zero.)

In the particular case of elements of the type \(g=g^0\), where \(g^0\) is an element to which the permutation \(P_g=1\) corresponds, from (6) we obtain

\[ X(g^0)=\sum_{\text{over sets}} X^{(\Gamma_1)}(g_1^0)\ldots X^{(\Gamma_N)}(g_N^0), \tag{7} \]

where \(g_1,g_2,\ldots,g_N\) are the “record” of the element \(g\) in each of the point groups \(G_1,G_2,\ldots,G_N\), respectively. In this case each of the ions of the complex forms a separate cycle.

If the group \(G_k\) is decomposed with respect to the invariant subgroup* \(G_k^J\) (which includes only elements of the type \(g^0\)),
\[ G_k=G_k^J+I_1G_k^J+\ldots+I_sG_k^J, \]
then each element \(g\in G_k\) can be represented in the form \(g=Ig^0\) (where \(I\) is an element not belonging to \(G_k^J\)), and formula (6) in the form

\[ X(Ig^0)=[\pi^n\ldots \bar\pi^{\bar n}] \sum_{\text{over sets}}^{\prime} X^{(\Gamma)}(I^mg_1^0g_2^0\ldots g_m^0)\ldots X^{(\bar\Gamma)}(I^{\bar m}\bar g_1^0\bar g_2^0\ldots \bar g_{\bar m}^0). \tag{8} \]

For a complex of two equivalent impurity ions (a pair), from (6)–(8) we obtain:

\[ X(g^0)= \begin{cases} X^{(\Gamma)}(g_1^0)X^{(\Gamma)}(g_1^0), & \text{if } a_1\Gamma_1=a_2\Gamma_2;\\[4pt] X^{(\Gamma_1)}(g_1^0)X^{(\Gamma_2)}(g_2^0) + X^{(\Gamma_2)}(g_1^0)X^{(\Gamma_1)}(g_2^0), & \text{if } a_1\Gamma_1\ne a_2\Gamma_2; \end{cases} \tag{9a} \]

\[ X(g)=X(Ig^0)= \begin{cases} (-1)^nX^{(\Gamma)}(g^2)=(-1)^nX^{(\Gamma)}(I^2g_1^0g_2^0), & \text{if } a_1\Gamma_1=a_2\Gamma_2,\\[4pt] 0, & \text{if } a_1\Gamma_1\ne a_2\Gamma_2. \end{cases} \tag{9b} \]

* For a pair of impurity ions in such a decomposition there will be two cosets, and in the general case more.

It is seen from (96) that in formula (3) of paper (¹), under the character sign, the quantity \(I^2\) was omitted. In the case when \(I^2 = Q\) (rotation by \(2\pi\)) (for example, for \(I = C_2\)), this is important for double-valued representations \(\Gamma\) (\(Q\) changes the sign of the character). In connection with this, the following corrections must be made in Table 1 of article (¹): \(\overline{2A} \times \overline{2A} \to 3A_1 + A_2\) (pair I) and \(3A + B\) (pair III), and also \(\overline{E} \times \overline{E} \to 2A_1 + E\) (pair I) and \(3A + B\) (pair III). In addition, the misprint \(\overline{E} \times \overline{E} \to A_g + A_u + E_u\) (pair V) should be corrected.

Examples of determining the states of complexes with \(N > 2\) will be considered by the authors separately.

We note that the proposed method is also applicable to free polyatomic molecules if one sets \(G_C = R_C\) and \(G_i = R_i\) (\(R\) is the rotation group). In this case our method differs from that proposed by Kotani (²) in that the spin–orbit interaction is taken into account from the very beginning.

Institute of Metal Physics
Academy of Sciences of the USSR

Ural State University
named after A. M. Gorky

Institute of Metallurgy, Sverdlovsk

Received
24 II 1967

CITED LITERATURE

¹ S. V. Vonsovskii, V. I. Cherepanov et al., DAN, 170, No. 6, 1288 (1966).
² M. Kotani, Proc. Phys.-Math. Soc. Japan, 19, No. 5, 460 (1937).