C O M P U T E R S I M U L A T I O N O F T H E O N S E T O F A F I R S T O R D E R M A G N E T I C INSTABILITY IN A FINITE MODEL SYSTEM W. KWO Department of Physics, Garyounis University, Baida, Libya
The variation with interatomic distance of one-electron and electron interaction energy differences of the low-lying energy levels of the H12 ring is related to the onset of a first order magnetic instability in terms of numerical results obtained in the projected Hartree-Fock approximation.
In studying magnetic aspects of changes in localization of electrons in periodic structures, the Coulomb interaction between electrons has received considerable emphasis . In this connection, we present herein numerical results obtained for a ring of twelve equidistant hydrogen atoms (H12), with each atom contributing a single ls electron. Unlike closed-sheU rings of 4n + 2 (n = 0, 1. . . . ) equidistant hydrogen atoms (see, for example, ref.  and references cited therein for discussions regarding the n = 0 case), the nonclosed shell nature of this orbitally degenerate ring of 4n (n = 3) equidistant hydrogen atoms yields the possibility of obtaining a ground state that changes multiplicity as the interatomic distance R is varied . In what follows, we consider the role of the interelectronic Coulomb repulsion energy in the onset of this transition within the context of symmetry adapted total wave functions which are pure spin states obtained by applying spin projection operators to states formed as in the unrestricted H a r t r e e - F o c k method. This permits electrons of opposite spin to avoid each other, and the total energy E of the system has the correct asymptotic behavior at large R. Let us first assume that the Hamiltonian H of the system is of the form
x a L o a L o ak , o
In (1), T is a one-electron operator that includes the kinetic energy of the electrons, the periodic potential of the hydrogen nuclei, and the Coulomb repulsion energy of these nuclei. The operator V represents the Coulomb interaction between the electrons eZ/r12. The creation and annihilation op-
erators a m~+ o and am,¢1 (m = k 1, k2, k'1, k~) are for electrons in Bloch states Ira, o) given by (1) of ref. [21. Introducing operators A~-(m) and B~-(m) as in eq. (2) of ref.  corresponding to states obtained in terms of a mixing of angle 0 in the space spanned by a m+, o and a --(2n--m), + o, we find that with 0 -- 0 and ~r/4 electrons occupy Bloch states and states similar to that of fig. 5.7 of ref. , respectively. The total wave functions I~b) so determined has the property that the overemphasis of the instability of the paramagnetic 0 = 0 state relative to the 0 > 0 state (implying a form of order leading to antiferromagnetic spin structure in the case of the sing,let state of long polyenes ) characteristic of the random phase and extended H a r t r e e - F o c k approximations is reduced . With H 0 and H i representing the first and second terms of (1) respectively, the one-electron energy E o and the electron interaction energy E i are given by E o = @ l H 0 / ~ ) / ( ~ l ~ ) and E i = (~lHil~)/(~lq~). The optimized values of 0 used in evaluating E o and E i w e r e obtained through minimizing the total energy E = <~lnl~>/@l~> for various R. In these calculations, the interactions between all neighbors were taken into consideration as in ref.  and 0 was varied in 1 min intervals in the vicinity of the interatomic distance R c where the ground state changes multiplicity. Let us now denote the one-electron energies E 0 of the low-lying 31"2 and IF4 states by E 0 (3F2) and Eo(IF4). With a similar notation for El, we define the energy differences AE 0 -. Eo(aF2)- Eo(lF4) and AE i ~-- E i ( a F 2 ) - Ei(IF4). In table 1, we give selected values of AE 0 and A E i calculated for 1.00 au < R < 2.75 au (even though this paper is concerned primarily with the behavior of AE o and AE i for 1.00 au < R < 2.75 au, it should be mentioned
Journal of Magnetism and Magnetic Materials 15-18 (1980) 875-876 @North Holland
IV. Kwo / Computer simulation of a magnetic instability
that we have obtained calculated results for large R showing that AE i + AE 0 tends to zero as R exceeds 5 au). These results show that AE 0 < 0 and AE i > 0 over this range of R, implying that in the absence of H i, the ground state is the 3F 2 triplet state. In the presence of H i, the ground state remains the 3F2 state at R --- 1.00 au because there AE i is less than four fifths of - A E o. As R increases from 1.00 to 2.00 au, the increase in AE i is approximately twice the corresponding decrease in AE o. Thus, in order to avoid the relatively larger electron interaction energy of the 31"2 triplet state, the system undergoes a spontaneous transition to the singlet ~F4 state at a lattice spacing R c that is found to be approximately 1.1 au. Since the magnetic m o m e n t of this finite system (which m a y be regarded as an order parameter) changes discontinuously at Re, we also conclude that this transition is a first order one. The author wishes to thank Professor P. R. Fulde for a very helpful discussion. The use of the computing facilities of G a r y o u n i s University (formerly the University of Benghazi) is acknowledged.
TABLE 1 Energy differences AEo and AE i as a function of interatomic distance R for the Hi2 ring R
1.00 2.00 2.75
-0.0261464 -0.0950783 -0.1202762
0.0207004 0.1696622 0.2955354
References  B. H. Brandow, Advan. Phys. 26 (1977) 651.  W. T. Kwo, in: Proc. Intern. Conf. Magnetism 1973, vol. 2 (Nauka, Moscow, 1974) p. 270.  N. F. Mott and E. A. Davis, Electronic Processes in NonCrystalline Materials (Clarendon Press, Oxford, 1971) p. 136.  I. A. Misurkin and A. A. Ovchinnikov, Mol. Phys. 27 (1974) 237.  B. Johansson, Ann. Phys. 74 (1972) 369.