UDC 538.1

PHYSICS

F. I. MUFTAKHOVA, I. G. KAPLAN, A. N. MEN’

ON THE ACCOUNT OF ORBITAL STATES IN THE THEORY OF INDIRECT EXCHANGE

(Presented by Academician S. V. Vonsovskii, 31 III 1967)

There exists a large class of magnetically ordered crystals in which paramagnetic ions are surrounded by nonmagnetic ions, for example crystals of the MnO type. The magnetic properties of this type of crystal may be described by the magnetic properties of the linear chain Mn—O—Mn*. For the first time such a model was considered by Kramers \((^{1})\), Anderson \((^{2})\), and Shimizu \((^{3})\). In work \((^{3})\) it was shown that the effective exchange integral is a quantity of third order of smallness in the parameter \((E_0-E_1)^{-1}\), if virtual transitions of one of the anion electrons to the unoccupied \(s\)-orbital of the cation are considered as the excited state (here \(E_0\) and \(E_1\) are the average energies of the ground and excited states, respectively)

\[ I_{\text{eff}}=-4\rho^2(I_{pd}-I_{ds})I_{pd}/(E_0-E_1)^3, \tag{1} \]

where \(\rho\) is the one-particle transfer integral,

\[ \rho=\int \Psi_s^*(r_1)\hat H_1(r_1)\Psi_d(r_1)\,d\tau,\quad I_{pd}=\int \Psi_p^*(r_1)\Psi_d^*(r_2)\frac{1}{r_2}\Psi_p(r_2)\Psi_d(r_1)\,d\tau, \]

\[ I_{sd}=\int \Psi_s^*(r_1)\Psi_d^*(r_2)\frac{1}{r_{12}}\Psi_s(r_2)\Psi_d(r_1)\,d\tau, \]

\(s,d\) denote the empty and singly occupied orbitals of the cation, and \(p\) is the occupied orbital of the anion.

In the present note an attempt is made to take into account the orbital states of the electrons within the same model of a linear chain of 3 centers—4 electrons. The purely ionic configuration is described by the antisymmetric wave function \(|[\lambda]k;\Gamma_1\mu_1[\lambda_1]\Gamma_2\mu_2[\lambda_2]\Gamma_3\mu_3[\lambda_3]\rangle\), where \([\lambda]\) is an irreducible representation of the permutation group of 4 electrons, \([\lambda]=[2^2]\) for the singlet state of the chain and \([\lambda]=[21^2]\) for the triplet; \(k\) is the Yamanouchi symbol; \(\Gamma_1\) is an irreducible representation of the cubic group obtained from the \(d\)-state of the electron, \(\mu_1\) is a row of this representation; \(\Gamma_2\) is an irreducible representation of the cubic group obtained from the \((p^2\,{}^1S)\)-state of the anion, \(\mu_2\) is a row of this representation. The excited configurations, analogously to Anderson et al., are considered as transitions of one electron of the anion to the cation, the transferred electron being placed on the unoccupied \(s\)-orbital. The wave functions of the excited configurations have the form: \(|[\lambda]k;\Gamma_1\mu_1[\lambda_1]\Gamma_2\mu_2[\lambda_2]\Gamma_3s\mu_1[\lambda_3]\rangle\) (the notation is analogous).

The wave functions of the ground and excited configurations may be represented in the form

\[ |[\lambda]k;\Gamma_1\mu_1[\lambda_1]\Gamma_2\mu_2[\lambda_2]\Gamma_3\mu_3[\lambda_3]\rangle = \]

\[ = \sum_{M_1M_2M_3} P^{\mu_1\mu_2\mu_3 k}_{M_1M_2M_3} |[\lambda]k;\,L_1M_1[\lambda_1]L_2M_2[\lambda_2]L_3M_3[\lambda_3]\rangle^{**}, \]

where \(P^{\mu_1\mu_2\mu_3}_{M_1M_2M_3}\) is the matrix of the projection operator \((^{5})\).

* The available neutron-diffraction experiment to some extent justifies this approximation \((^{4})\).

** For the ground configuration only the functions of the \(d\)-electron formed according to the \(e_g\)-representation are taken into account, and for the excited configurations (the \((sd)\)-configuration) only functions of \(E_g\)-symmetry.

Notation: \(A_1=E_0-\frac{2}{3}I_{pd}^{02}-\frac{4}{3}I_{pd}^{12}-E;\quad A_4=E_0-\frac{2}{3}I_{pd}^{00}-\frac{4}{3}I_{pd}^{10};\)

\[ A_2=A_3=E_0-\frac{1}{3}I_{pd}^{00}-\frac{1}{3}I_{pd}^{10}-\frac{2}{3}I_{pd}^{02}-\frac{2}{3}I_{pd}^{12}. \]

For singlet states:

\[ B_m= \left| \begin{array}{cc} E_1+I_{sd}^{02}+\frac12 I_{pd}^{m2}-\frac12 I_{ps}^{m0}-E & \frac12\sqrt3\,(I_{ps}^{m0}-I_{pd}^{m2}) \\[2mm] \frac12\sqrt3\,(I_{ps}^{m0}-I_{pd}^{m2}) & E_1-I_{sd}^{02}-\frac12 I_{pd}^{m2}+\frac12 I_{ps}^{m0}-E \end{array} \right|, \]

\[ C_m= \left| \begin{array}{cc} E_1+I_{sd}^{00}+I_{pd}^{m2}-\frac12 I_{pd}^{m2}-\frac12 I_{ps}^{m0}-E & \frac12\sqrt3\,(I_{ps}^{m0}-I_{pd}^{m0}) \\[2mm] \frac12\sqrt3\,(I_{ps}^{m0}-I_{pd}^{m0}) & E_1-I_{sd}^{00}-I_{pd}^{m2}+\frac12 I_{pd}^{m0}+\frac12 I_{ps}^{m0}-E \end{array} \right|, \]

\[ D_m= \left| \begin{array}{cc} E_1+I_{sd}^{02}+I_{pd}^{m0}-\frac12 I_{pd}^{m2}-\frac12 I_{ps}^{m0}-E & \frac12\sqrt3\,(I_{ps}^{m0}-I_{pd}^{m2}) \\[2mm] \frac12\sqrt3\,(I_{ps}^{m0}-I_{pd}^{m2}) & E_1-I_{sd}^{02}-I_{pd}^{m0}+\frac12 I_{pd}^{m2}+\frac12 I_{ps}^{m0}-E \end{array} \right|, \]

\[ F_m= \left| \begin{array}{cc} E_1+I_{sd}^{00}+\frac12 I_{pd}^{m0}-\frac12 I_{ps}^{m0}-E & \frac12\sqrt3\,(I_{ps}^{m0}-I_{pd}^{m0}) \\[2mm] \frac12\sqrt3\,(I_{ps}^{m0}-I_{pd}^{m0}) & E_1-I_{sd}^{00}-\frac12 I_{pd}^{m0}+I_{ps}^{m0}-E \end{array} \right|, \]

\[ L= \left| \begin{array}{cc} \frac12\sqrt2\,\widetilde I_{pd} & 0\\ 0 & -\frac12\sqrt2\,\widetilde I_{pd} \end{array} \right|, \qquad K= \left| \begin{array}{cc} -\frac14\sqrt2\,\widetilde I_{pd} & -\frac14\sqrt6\,\widetilde I_{pd}\\ -\frac14\sqrt6\,\widetilde I_{pd} & \frac14\sqrt2\,\widetilde I_{pd} \end{array} \right|, \]

\[ \widetilde I_{pd}= \int \Psi_{d2}^{*}(r_1)\Psi_{p-1}^{*}(r_2) \frac1{r_{12}} \Psi_{d0}(r_2)\Psi_{p1}(r_1)\,d\tau . \]

For triplet states:

\[ B_m= \left| \begin{array}{ccc} E_1+I_{sd}^{02}-\frac32 I_{pd}^{m2}-\frac12 I_{sp}^{0m}-E & \frac12\sqrt3\,(I_{sp}^{0m}-I_{pd}^{m2}) & 0 \\[2mm] \frac12\sqrt3\,(I_{sp}^{0m}-I_{pd}^{m2}) & E_1-I_{sd}^{02}+\frac16 I_{pd}^{m2}+\frac12 I_{sp}^{0m}-E & \frac23\sqrt2\,I_{pd}^{m2} \\[2mm] 0 & \frac23\sqrt2\,I_{pd}^{m2} & E_1-I_{sd}^{02}-\frac23 I_{pd}^{m2}-I_{sp}^{0m}-E \end{array} \right|, \]

\[ C_m= \left| \begin{array}{ccc} E_1+2I_{sd}^{02}-I_{pd}^{m2}-\frac12 I_{sp}^{0m}-\frac12 I_{pd}^{m2}-E & \frac12\sqrt3\,(I_{sp}^{0m}-I_{pd}^{m0}) & 0 \\[2mm] \frac12\sqrt3\,(I_{sp}^{0m}-I_{pd}^{m0}) & E_1-I_{sd}^{02}+\frac12 I_{pd}^{m0}+\frac12 I_{sp}^{0m}-\frac23\sqrt2\,I_{pd}^{m2} -\frac13 I_{pd}^{m2}-E & \frac23\sqrt2\,I_{pd}^{m2} \\[2mm] 0 & \frac23\sqrt2\,I_{pd}^{m2} & E_1-I_{sd}^{00}-I_{pd}^{00}-I_{sd}^{0m}+\frac13 I_{pd}^{m2}-E \end{array} \right|, \]

\[ D_m= \left| \begin{array}{ccc} E_1+I_{sd}^{02}-I_{pd}^{m0}-\frac12 I_{sp}^{0m}-\frac12 I_{pd}^{m2}-E & \frac12\sqrt3\,(I_{sp}^{0m}-I_{pd}^{m2}) & 0 \\[2mm] \frac12\sqrt3\,(I_{sp}^{0m}-I_{pd}^{m2}) & E_1-I_{sd}^{02}+\frac12 I_{pd}^{m2}+\frac23\sqrt2\,I_{pd}^{m0} +\frac12 I_{sp}^{0m}-E & \frac23\sqrt2\,I_{pd}^{m0} \\[2mm] 0 & \frac23\sqrt2\,I_{pd}^{m0} & E_1-I_{sd}^{00}-\frac23 I_{pd}^{0m}-I_{sp}^{0m}-E \end{array} \right|, \]

\[ F_m= \left| \begin{array}{ccc} E_1+I_{sd}^{02}-\frac32 I_{pd}^{m0}-\frac12\sqrt3\,(I_{sp}^{0m}-I_{pd}^{m0}) -\frac12 I_{sp}^{0m}-E & 0 & 0 \\[2mm] \frac12\sqrt3\,(I_{sp}^{0m}-I_{pd}^{m0}) & E_1-I_{sd}^{00}+\frac16 I_{pd}^{m0}+\frac23\sqrt2\,I_{pd}^{m0} +\frac12 I_{sp}^{0m}-E & \frac23\sqrt2\,I_{pd}^{m0} \\[2mm] 0 & \frac23\sqrt2\,I_{pd}^{m0} & E_1-I_{sd}^{00}-\frac23 I_{pd}^{m0}-I_{sp}^{0m}-E \end{array} \right|, \]

\[ L= \left| \begin{array}{ccc} -\frac14\sqrt2\,\widetilde I_{pd} & -\frac14\sqrt6\,\widetilde I_{pd} & 0\\ -\frac14\sqrt6\,\widetilde I_{pd} & \frac12\sqrt2\,\widetilde I_{pd} & 0\\ 0 & 0 & -\frac12\sqrt2\,\widetilde I_{pd} \end{array} \right|, \]

\[ K= \left| \begin{array}{ccc} -\frac12\sqrt2\,\widetilde I_{pd} & 0 & 0\\ 0 & -\frac16\sqrt2\,\widetilde I_{pd} & \frac23\widetilde I_{pd}\\ 0 & \frac23\widetilde I_{pd} & \frac16\sqrt2\,\widetilde I_{pd} \end{array} \right|. \]

\[ \alpha=\frac16\sqrt6\,\rho_1,\qquad \beta=\frac12\sqrt2\,\rho_1,\qquad \gamma=\frac16\sqrt6\,\rho_2,\qquad \delta=\frac12\sqrt2\,\rho_2. \]

\(\sigma=0\) for triplet states; for singlet states, the rows and columns containing \(\sigma\) need not be taken into account.

Table 1

Visible off-diagonal stepped blocks in the surrounding part of the table are labeled \(K\) and \(L\).

For the calculation of the energy matrix, the method of genealogical coefficients was used (⁶). Without dwelling on the details of the calculation, we give the energy matrix (see Table 1). The form of the energy matrix is the same both for the triplet and for the singlet states (for the singlet the order of the matrix is 52, for the triplet 76).

We do not set ourselves the task of finding all the roots of the energy matrix; rather, as previous authors did, we seek only the lowest levels by Löwdin’s method (⁷). The solution averaged over the orbital states of the cations has the form:

\[ I_{\mathrm{eff}}=\frac{E_s-E_t}{2} =-\frac{1}{3}(E_0-E_1)^{-3}\{\rho_1^2[I_{pd}^{02}(I_{pd}^{02}+I_{pd}^{00}-2I_{sd}^{02})+ \]

\[ +I_{pd}^{00}(I_{pd}^{00}+I_{pd}^{02}-2I_{sd}^{00})]+ \]

\[ +\rho_2^2[I_{pd}^{12}(I_{pd}^{12}+I_{pd}^{10}-2I_{sd}^{02})+ I_{pd}^{10}(I_{pd}^{10}+I_{pd}^{12}-2I_{sd}^{00})]\}, \tag{2} \]

where \(I_{pd}^{\alpha\beta}\), \(I_{sd}^{\alpha\beta}\) have the same meaning as in formula (1), and the upper indices denote projections of the orbital moments, for example,

\[ I_{pd}^{12}=\int \Psi_{p1}^{*}(r_1)\Psi_{d2}^{*}(r_2)\frac{1}{r_{12}}\Psi_{p1}(r_2)\Psi_{d2}(r_1)\,d\tau; \]

\[ \rho_1=\int \Psi_s^{*}(r_1)\hat H_1(r_1)\Psi_{p0}(r_1)\,d\tau,\qquad \rho_2=\int \Psi_s^{*}(r_1)\hat H_1\Psi_{p1}(r_1)\,d\tau. \]

If in formula (2) we assume that all \(I_{pd}^{\alpha\beta}=I_{pd}^{00}\), \(I_{sd}^{\alpha\beta}=I_{sd}^{00}\), \(\rho_1=\rho_2\), then we obtain the formula

\[ I_{\mathrm{eff}}=-\frac{4\rho_1^2(I_{pd}^{00}-I_{sd}^{00})I_{pd}^{00}}{(E_0-E_1)^3}, \]

which coincides with (1), if \(I_{pd}\), \(I_{sd}\), and \(\rho\) are understood as \(I_{pd}^{00}\), \(I_{sd}^{00}\), and \(\rho_1\). The formula obtained, (2), makes it possible to carry out numerical estimates of the effective exchange interaction as a function of the orbital states.

In conclusion we take the opportunity to express our gratitude to S. V. Vonsovskii and B. V. Karpenko for a useful discussion.

Ural State University
named after A. M. Gorky

Institute of Metallurgy, Sverdlovsk

Physicochemical Institute
named after L. Ya. Karpov

Received
17 III 1967

CITED LITERATURE

¹ H. A. Kramers, Physica, 1, 182 (1934).
² P. W. Anderson, Phys. Rev., 79, 350 (1950).
³ M. Shimizu, Progr. Theor. Phys., 8, 416 (1952).
⁴ G. W. Pratt, Phys. Rev., 97, 926 (1955).
⁵ J. C. Slater, Electronic Structure of Molecules, Moscow, 1966.
⁶ I. G. Kaplan, ZhETF, 41, 790 (1961); Theor. Exp. Chem., 3, No. 2 (1967).
⁷ Löwdin, J. Chem. Phys., 19, 1396 (1951).