Volume 115, number 6 CIIEMICAL PHYSICS LETTERS I9 April 1985

ON -l-DE UNIVERSAL INSTABILITY OF THE OPEN-SIPELL BBSTRI~D I-IABmE-FOCK METHOD

M.M. MESTECHKIN, G-T. KLIMKO and G.E. WHYMAN

Insmute of Physrcal Organic and Coal Chemisny, R Luxembaug SL 70, 340114 Donersk, USSR

Received 11 April 1984; in final form 15 January 1985

The open-shell restricted Hartee-Fock (RHF) method is shown to be unstable with respect to spin-fhpping variations for arbitrary spin s f 0. Introducing the spin-purity restriction on the wavefunction variation removes this universal instability_

1_ introduction

There are numerous indications of the connection between the Hartree-Fock instability and real chemi- cal phenomena [ 1,2]. However, the origin of this con- nection still remains somewhat vague [3]. Perhaps, this may be partially attributed to a certain incom- pleteness of open-shell Hartree-Fock stability theory itself_ The sotalied spin-flipping instability [4], re- sulting from the opposite-spin variations of the occu- pied MQs, presents one example. Although a conjec- ture on its general nature has been made from numeri- cal experience in the case of doublet and triplet states [2], a strict proof is still lacking. The reason is that the spin-flipping instability arises only in second order in orbital deviations, in contrast to the more common first-order different-orbitals for different-spins (DODS) instability, and one usually believes that in this situa- tion it would be necessary to solve the corresponding eigenvalue problem. However this is not the case, as will be shown in section 3 where the universal RHF spin-flipping instability is proved, for arbitrary spin s f 0, directly from the expression of the second ener- gy variation.

Clearly, this universal instability does not depend on the particular system considered and therefore can- not give any physical information. lt will be eliminated in section 4 by introducing the spin-purity restriction on a ~ariiztion of the total wavefimction. The density matrix formalism briefly outlined for the open-shell

RHF method in section 2 is used throughout, Due to its invariance, this formalism is especially useful in the open-shell case since it allows simultaneous considera- tion of the variations of doubly and singly occupied orbitals. Its additional advantage lies in the simplicity and brevity of the derivation.

2. BHF density matrix and energy variations

Our starting point is the standard RHF-type deter- minant built of doubly and singly occupied orbitals. The corresponding density matrix, when spin factors used are pure a and P, is well known [5] :

%-IF = m+ y3ql +-qi 3 (1)

where I + Y and Z are the spin-free density matrix aud the spin-density matrix, respectively, I is rhe unit ma- trix, and u are the Pauli matrices. The convenience of the Y matrix is that the above restrictions on orbitals leave Y as the only independent quantity [6,7]:

Z=I-- Y2, Y=Y3. (2)

30th relations can be easily verified in the MO basis where Y and Z are diagonal and

Y*i = 1, Ykk=o, Y7=-1, Z,,=l,

Zii = Zjj = 0 . (3)

The eigenvalues of Yin (3) correspond to the occupa-

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Volume 115, number 5 CHEMICAL PIiYSICS LET-TERS 19 April 1985

numbers 2 , ! , 0 of doubly occupied, singly occu- pied and empty I i0.s enumerated by the indices i, k, i respectively.

Starting with the RHF wavefunction, we shall con- sider the most general variations that violate the RHF property but conserve the single-determinant form. In terms of the density matrix variation 6P this means that

G~=R,-,o, +Ozaz +Da+ +D+a_ , (4)

but p* = 9 and therefore [3’, 691, = 0. The latter limitation can be reformulated for the spinless compo- nents of 6 3 as [8.9]

~~~4d, + Lc 41, = cy, &I, + E Dal+ = 0 3 (5)

YLp+DYp=o. (6)

by using eqs. (1) and (4) and the well-known properties of the Pauli matrices. Here the matrices DO. LJ, are Ezrmitean while D may be non-Hermitean and Df is the Hermitean conjugate of D. For the sake of brevity we continue to use the symbol 2 for I - r;? and intro- duceY,=Y+Z,Yg=Y-2.

The matrices Do, Dz correspond to the usual DODS variation of MOs while the matrix D corresponds to the spin-flipping variation mixing the MOs of cx and 0 spin. As is seen from eqs. (5) and (6) these two kinds of dis- tortion of the RHF wavefunction are independent from each other and can be treated separately_ This property also re_mains true for energy variations_ The energy of the single-determinant wavefunction built of general spin orbitals is expressed through the density matrix 9 with the spin structure similar to that in (4)

E=Try[CNsO +sL(‘P)] _ (7)

For energy variations it is readily seen that

6E=Tr63[Noo +2z(3)3, (8)

s2E=TrC62y[WOO+22(3’)] +2i?iyda@T)) _ (9)

In eqs. (7)-(g) z symbolizes the averaged interelec- tron-interaction operator in spin-orbital form_ It can be expressed through the usual Coulombexchange, G, and exchange, K, operators [S] _ For example,

2 St (S 9) = G(Do) o. + K(o,) o, + K(D) cr,

+W~J_ , (10)

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where G(A) and K(A) are presented in the orthogonal basis as

Gp&J) = Kp&) + c WI WA,, , (11) St

for an arbitrary matrix A.

3. Universal instability of open-shell RHP solution

The basic relations in our treatment of the open- shell RHF stability problem are eqs. (8) and (9) where 3’ must be set equal to 3- of eq. (1). It then fol- lows that only the DODS density matrix variations, Do, O,, contribute to SE. By putting Do = 0 it may be shown that 6E = 2 Sp D,K(Z) where the symbol Sp, unlike Tr, does not include a summation over the spin variables_ One can further choose D, = -KX, with K a small positive quantity with dimensions of (energy)-l and

x= Y%(Z)Y2 - YK(Z)Y _ (12)

The energy is thus decreased, 6E = -K Sp X2 < 0, and we have shown simultaneously the validity of the nec- essary limitation (5) on D, since [Y, Xl + = ZX = XZ = 0. This result [lo] extends to arbitrary non-zero spin proof [ 11 J of the RHF solution instability relative to the DODS state, previously obtained for doublets. In particular one obtains with the aid of eqs_ (3), (11) and (12) the natural generalization of the eq. (4) in ref. [43

2s

Xii = - kq, (ik I7@ _ (13) *

FOT spir+~pping variations, the only non-zero ma- trix in eq. (4) is D. Then 6E vanishes, and the second energy variation 8*E is needed to solve the stability problem. After the spin reduction, eq. (9) may be cast in the form

S&E = 2 Sp D+ I-&D - Di$ + 2K(D)] ,

N

(14)

%B =;[Fo’K(Z), YaJ+, Fo=U+G(l.+Y).

When deriving eq. (14) the condition (6) and the rela- tion s29 = 2(6p)2 - 43@ sPj2 are taken into account,

Volume 115. number 6 CHEMICAL PHYSICS LETTERS 19 April 1985

The latter is valid, as can be verified for the Fock- Dirac matrix 3. when the spin-orbital variations are orthogonal to occupied spin-orbitals. But even under this restriction, it is now shown that the RHF solution is always unstable to spin-flipping. In other words one may choose some particuhuD for which 6&!? < 0. We put D = aZ + KX with Q being a small dimensionless number, and fmd

@jE = 6;E + 2QK sp x2 , (15)

on account of the identities YZ = 2X = XZ = [Y, X’J+ = 0 following from eqs. (2) and (12). Clearly, if X # 0 then 6&7 < 0 for Q < -&E/2 Sp(KX)* (to first order in both a and Kj. Q.E.D.

The remainin g possibility X = 0 admits a weaker in- equality min g&5 d 0 = S&E_ Due to symmetry re- strictions, the pathological case, X = 0, sometimes emerges in actual calculations [1,4] as a result of em- ploying a minimal basis set. Sufficient extension of the basis will always lead to the appearance of an empty orbital,j, having the same symmetry as some doubly occupied orbital, i. Then, according to eq. (13), X f 0 and 6*E < 0. The analogous change in the symmetry of levels may also occur in the minimal basis at the point of the orbitalenergy crossing [l] where an empty orbitalj in eq. (13) becomes symmetric and a non-zero integral <ik l&0 appears. As a result both the DODS and spin-flipping instabilities of the BHF wavefunction arise simultaneou.sIy once the matrix X diverges from zero.

4. Stability under spin symmetry restriction

In the RHF case it is natural to classify spin-flipping variations D according not only to a change AM in the

total-spin projection of the wavefunction variation 6* but also to the value of total spin. Analysis shows that the appropriate limitations on the D matrix are, for AM = -1. and arbitrary AS,

Y$=D=-DYP, (16)

whereas, for fUU = As = 1,

D=-Y$=-YpD=DYa=DYg,

andforAM=As=-1

D=ZD=DZ, SpD=O.

(17)

(18)

The above proof of the inst&‘&ty can be transfered to the case of LyM= -1. Indeed, the derivation of the inequality S2E < 0 here follows that of section 3 in detail if the trial matrix D is chosen in the form a2 + K(X + Yx) satisfying eq. (16). Consequently, the spin-flipping instability depends on the admixture of functions of arbitrary As and AM = - 1 in 6q. Th_e key part in this admixture is played by the function S_\ko belonging to the same multiplet as qo_ Since ah func- tions of some multiplet are degenerate, one can guar- antee at least s2E = 0. The same conclusion in terms of the above used matrices follows frorr *he identity 6$E= O_ Note that the spin density matnx Z is also a spatial part of the transition density matrix between ‘kn and .!?_\kO. The other basic quantity X resulting from the distortion of the closed shell provides the strict inequality S2E < 0.

In order to clarify the origin of the established in- stability, we shall consider a simple threeelectron ex- ample_ Let the RHF function \ku = det ICY, X& cpar]/ 61i2 turn, after spin-flipping variation, into

*k= det[Xar+rr98,x& (PLY+&$]/[~(~ +a2)(1+b2)11j2,

where X, cp, JI are some orthogonal orbitals. The com- ponent 68, that is linear in a and 5, of * contains

g-q,-, = det [xar, xp, 1p/3]/6l/~ and 9, = det [r,!.$, x/3, ~0or]/6~/~. In the energy expression E = (\kp181 these terms interact with each other, but not with QO for the spinless Hamiltonian fi_ Thus, to the lowest order in a, b

E = E,, + a2(El - Eo) - Qb’&.$fX)) ,

where Eo = Cf?OIHl*O), E1 = (Q1lfiiQ1). ClearIy, one obtains E <E. by a suitable chaise of the small num- bers a and b_ It is seen that simultaneous variation of the singly and doubly occupied MC& is an essential feature of our proof, whereas in the existing derivation [ 121 of the secular spin-flipping instability equations such variations are considered separately.

However our present, proof cannot be transfered to matrices D that obey eqs. (17) and (18) because these limitations exclude D = a2 from the variety of admis- sible matrices as a consequence of the properties Yz =Z, Sp Z = 2s (see eqs. (2) and (3)). As a result, in these cases we must solve the eigenvalue problems, that are associated with (14)

-N$-DDNg+$(I- Y,)K(D)(I+Y$=A+D, (19)

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Volume 115, number 6 CHEMICAL PHYSICS LETTERS 19 April 1985

-z pq2-J + Fo]D - D [K(z) - F*J z + 2ZiqD)Z

=X-D, (20)

for As = AM = 1, and AM = As = -1, respectively. The ieft-hand sides of these equations are just projections of the term in brackets in eq. (14) onto subspaces of D matrices satisfying the eqs. (17) and (18). FinaIly it should be emphasized that in these cases of the squared-total-spin conservation, solving the StabiIity problem becomes equivalent to determining the low- est transition energy in the basis of all singly excited configurations. Indeed, the difference between the two eigenvalue problems is due to the matrix element (6 2\k I&I %i?$ entemg 6 2E. Doubly excited configura- tions forming 6 2\k may correspond to either AM = -C2 or AM = 0 by definition of spin-flipping excitations. The former do not interact with \kO while the latter do not appear in s2* at all when the restriction AM = +I is imposed on 6\k.

5. Discussion

Every approximate solution of the S&r&linger equation is “unstable” with respect to a more accurate enc. Th s trivial remark shows that instability of this kLd cannot give specific information on a particular Ffstem. As has been demonstrated, all variants of the kstability of the above type in the case of RI-IF are connected with the single non-zero matrix X (eq. (12)) which determines the direction and also the shift value [13] from RHF to a more accurate UHF wavefunction and simultaneously to a still more precise spin-ex- tended one, describing the same state. Therefore, in the framework of RHF it is natural to exclude these external instabilities from consideration. Then the only stability equations (19) and (20) and the internal FWF -. &ability equation (65) from ref. [lo] remain_ All of them guarantee spin conservation and consequently may indicate the actuaI intersection of potential hypersurfaces of different states with the same or dif-

ferent multiplicity. In the latter case the lowest instabil- ity eigenvalue coincides with the energy separation from an adjacent low-lying hypersurface calculated by means of the superposition of all singly excited configu- rations.

Thus, after elimination of the universal instability inherent in any approximate solution, the description of chemical phenomena in terms of the instability theory is reduced, in fact, to t&e determination of the energy level crossing. Probably, the most interesting problem of the appearance of new states conditioned by the interelectron interaction may be investigated within the framework of the RJ?lF method, using the consideration of its internal-instability threshold as well as in the closed-shell case [ 14,151.

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