J. Phys. Chem. 1982, 86. 1204-1212
1204
been done, by letting the protons also diffuse, but with a diffusion coefficient smaller by a factor of the ratio of the electron to proton mass. The fixed-node calculations12for the potential surface of Liz and LiH have not yet been carried as far as for He. The best variational Monte Carlo calculation for the equilibrium configuration using either Slater or Gaussian orbitals in the determinant, multiplied by a simple Pad6 approximant form of the pair wave function, yields about half the correlation energy.I3 (The correlation energy is the difference between the experimental result and the best Hartree-Fock calculation.) The subsequent fixednode calculations brings the results, within the rather large statistical error bar of about lo00 K, into agreement with experiments. Hence longer runs need to be made to bring the error bar down, so that the effect of the incorrect nodes can be ascertained. It is, however, already clear that the fixed-node approximation calculates in excess of 95% of the correlation energy in these rather few electron systems. With respect to the accuracy of obtaining potential surfaces the possibility should be pointed out that by using a differential Monte Carlo method, potential energy differences of neighboring regions on the potential surface can probably be calculated more accurately than the absolute energies by an order of magnitude. Finally, the preliminary results for the binding energy of a single molecule of water should be cited as repre-
sentative of what might be done for more complex molecules.'2 Again the variational Monte Carlo results can be considerably improved, since the importance functions used so far yield energies no better than the Hartree-Fock calculations. In general it is hoped that the variational Monte Carlo calculation should yield 90% of the correlation energy and the subsequent stochastic processes obtain 99.9% of the correlation energy. The fixed-node calculation for water, with a crude importance function, gives 80% of the correlation energy. The error bar is such that the remaining 20% is primarily due to the poor nodes of the importance function. Nevertheless, the accuracy of this preliminary calculation already exceeds the accuracy of the best CI calculation.16 This approach certainly represents a new direction for the theoretical quantum potential calculations, and at this stage it shows much promise.
Acknowledgment. We thank Professor J. Anderson for letting us quote his preliminary results on helium. This work was performed under the auspices of the U.S.Department of Energy by the Lawrence Livermore National Laboratory and Lawrence Berkeley Laboratory under contract No. W-7405-ENG-48and by the National Science Foundation under Contract No. CHE-7721305. (16) B. J. Roeenberg and I. Shavitt, J. Chem. Phys., 63, 2161 (1975); W. Meyer Znt. J . Quantum Chem., Symp. No. 5, 341 (1971).
Ordinary Fieid-Theoretic Methods for Self-Conslstent Wave Functions Which Describe Bond Formation and Dissociationt WaHer B. England Department of Chernlshy and The Laboratory fa? Surface Studies, The University of Wisconsin-Milwaukee, Milwaukee, Wlsconsin 5320 1 (Received: August 28, 198 1; In Final Form: January 12, 7982)
Three observations motivated the present work (1)the Bardeen-Cooper-Schrieffer (BCS) wave function may be used to describe chemical bonding in formally the same way it is used to account for Cooper pairing; (2) the BCS wave function and a spinor or quasi-particle representation induce the same field-theoreticmethods as the Hartree-Fock (HF) wave function and the usual second-quantization;and (3) a Lipkin Hamiltonian negates the manifest particle nonconservation of the BCS wave function. Self-consistent BCS wave functions and solution methods which are appropriate for chemical bonding are described in this paper. Orbitals are rigorously defined by even-replacement multiconfigurationalFock operators. Density distributions are defined by the Hartree-Bogoliubov equations. The Lipkin Hamiltonian is computed by successive approximations in a manner which is similar to the BCS and Nogami theories. Orbital equations which are effectively the same as the HF equations are also obtained.
1. Introduction
Self-consistent one-particle methods have proved useful for studying many-body systems. Any such method which employs a physical vacuum is amenable to the techniques of second quantization, namely, infinite order diagrammatic Feynman-Dyson perturbation, theory' and equations-of-motion (EOM) theory? It is usually true that the effectively one-particle methods violate known conserva'Presented, in part, at the 14th Midwest Theoretical Chemistry Conference, Chicago, May, 1981; the 36th Symposium on Molecular Spectroscopy, Columbus, June, 1981, Abstract RF8;and the International Symposium on New Directions in the Molecular Theory of Gases and Liquids, a tribute to Professor J. 0. Hirschfelder on his 70th birthday, Madison, June, 1981. 0022-365418212086-1204$0 1.2510
tion principles. Lipkid showed how to use vacuum wave functions which violate conservation principles. The Lipkin procedure employs model Hamiltonians, which I shall call Lipkin Hamiltonians. Self-consistent and/or field-theoretic methods may be based on the Lipkin Hamiltonians. The present work is concerned with particle nonconservation. Hartree-Fock (HF) theory leads naturally to the particle-hole vacuum, and vice versa. This vacuum is un(1) Mattuck, R. D. 'A Guide to Feynman Diagrams in the Many-Body Problem"; McGraw-Hill: New York, 1976; 2nd ed. (2) Rowe, D. J. "NuclearCollective Motion";Methuen: London, 1970; Part 11. (3) Lipkin, H. J. Ann. Phys. (N.Y.) 1960, 9, 272.
0 1982 American Chemical Society
Fiekl-Theoretic Methods
The Journal of Physical Chemistty, Vol. 86,No. 7, 1982 1205
suitable for describing correlations such as those that accompany phase transitions and the formation or rupture of chemical bonds. A more suitable vacuum appeared in the Bardeen-Cooper-Schrieffer4 (BCS) theory of superconductivity. The present paper applies generalized BCS methods to the chemical bonding process. The generalized fermion field variables for the BCS-type vacuum were reported by Bogoliubovs and Valatin.6 A detailed description of the self-consistent solution process was termed the generalized HF method by Valath~.~ The self-consistent theory is also called Hartree-Bogoliubov (HB) theoryS2 Infinite-order perturbation theories were reported by Gorkoq and N a m b ~ . ~The close formal connection between Nambu's description of HB theory and the usual HF theory is emphasized by Schrieffer.lo Nambu's spinor fields lead to the same diagrams that occur in the Feynman-Dyson perturbation theory of the particle-hole vacuum.lJo Earlier, Anderson" used the same spinor fields to describe the approximate EOM theory for the BCS vacuum. An extended HB-EOM formalism was given by Rowee2 Apparently, atomic nuclei are the only finite systems to which HB theory has been applied.2J2J3 Nogami and Zucker14 used the earlier method of Nogami15 to calculate an exact HB solution with a Lipkin Hamiltonian. The present paper derives a variant of HB theory which is appropriate for chemical bonds. 2. The BCS Description of Bonding The BCS method aims to calculate the cluster wave function
12n) = (ECk(2n)ak+a-k+)n'l) k>O
(1)
where the ak+are creation operators for the one-particle states C#Jk, and 1) is the bare vacuum. All single-particle states which have spin a (0)are labeled by k (4).The wave function (1)is used in pair correlation theories16and represents the system equally well for most values of the geometrical parameters. If there is one bond, then the simplest form for (1)is expressed as (2) = C1(2)l1,-l)
+ C2(2)(2,-2)
are two bonds, then the simplest form for (1)is
+
+ +
14) = C1(4)C2(4)ll,-1,2,-2) C1(4)C3(4)11,-1,3,-3) C1(4)C,(4)1l,-1,4,-4) + C,(4)C3(4)12,-2,3,-3) C,(4)C,"12,-2,4,-4) C3"C4"13,-3,4,-4)
+
(3)
using an obvious extension of notation. The wave function (3) is similar to the wave function used by Goddard et al.18 for systems with two bonds. Linderberg and Ohrn19have shown that the wave function (1)can represent the ground state of the self-consistent single-particle-hole propagator. As such, it is the self-consistent ground state of the random-phase approximation, and induces an excitation spectrum. This is indeed 'a very significant development, since it was previously not known how to consistently solve the random-phase problem. It is well-known that superconducting solutions may be less stable than HF solutions (see, e.g., section 11.6 of ref 2). In terms of the wave functions (1)which have n > 1, this means that certain of the Ck(2n)vanish. For example, suppose (3) is applied to the process
H ~ A H ~ + H ~ Further suppose that orbitals 1and 3 localize on one H2, and that orbitals 2 and 4 localize on the other. The wave function (3) will dissociate properly only if it reduces to the HF solution. In terms of the densities defined below, this means, for example, hl = h2 = 1,h3 = h4 = 0, which gives C1(4)= CJ4) = 1, CJ4) = CJ4) = 0. However, there is no a priori guarantee that the variational method will choose this solution; it is only guaranteed that the actual solution will never have an energy which is greater than the aforementioned HF solution. Thus, when n > 1, proper dissociation of (1)is not automatic. The BCS vacuum wave function is written as
I-)
=
n [(I - hk)l/' + (hk)'/'Uk+U_k+]-l)=
k>O
Projection of (4) onto the Hilbert space of 2n particles corresponds to (1). The projection is expressed as
(2)
where Im,-m) signifies the double occupancy of 4m. The wave function (2) was used by Das and W a h P and Goddard et al.18 to describe single bonds. Similarly, if there and hence (4)Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. Phys. Rev. 1967,108, 1175. ( 5 ) Bogoliubov, N. N. Nuovo Cim. 1958,7 (ser. lo),794. (6)Valatin, J. G. Nuovo Cim. 1958,7 (ser. lo),843. (7) Valatin, J. G. Phys. Rev. 1961, 122,1012. (8)Abrikosov, A. A.; Gorkov, K. P.; Dzyaloohinskii,I. Y. "Quantum Field Theoretical Methods in Statistical Physics"; Pergammon: New York, 1965; Chapter VII. (9)Nambu, Y. Phys. Reu. 1960,11 7 , 648. (10)Schrieffer,J. R."Theory of Superconductivity";Benjamin: New York, 1964; Chapter 5. (11)Anderson, P. W. Phys. Rev. 1958,112,1900. (12)Migdal, A. B. "Theory of Finite Fermi Sydtems"; Interscience: New York, 1967. (13)Fetter, A. L.;Walecka, J. D. "Quantum Theory of Manyparticle Systems"; McGraw-Hilk New York, 1971. (14)Nogami, Y.; Zucker, I. J. Nucl. Phys. 1964,613,203. (15)Nogami, Y. Phys. Rev. 1964,134,B313. (16)Kutzelnigg, W. "Modern Theoretical Chemistry"; Plenum: New York, 1977;Vol. 111, and references therein. (17)Wahl, A. C.;Das, G. "Methods of Electronic Structure Theory"; Plenum: New York, 1977;Vol. 111, Chapter 3. (18)Hung, W. J.;Hay, P. J.; Goddard, W. A. J.Chem. Phys. 1972,57, 738.
Because (4) does not conserve particles, BCS theory typically supplies saddle point approximations to (1).20 However, a Lipkin Hamiltonian may be used to obtain (1) exactly. The method proposed by Weiner and Goscinski,2l which is based on a direct optimization method,= also may be used to compute (1)exactly. This approach appears to be promising when the summation over k and the number of particles 2n combine to give a large number of terms in (1). It is worth emphasizing that any practical (19)Linderberg, J.; Ohm, Y. Int. J. Quantum Chem. 1977,12, 161; 1979,15,343. (20)Bayman, B. F. Nucl. Phys. 1960,15,22. (21)Weiner, B.; Goscinski, 0. Phys. Rev. A 1980,22, 2374;Int. J. Quantum Chem. 1980,17,1109; Phys. Scr. 1980,21,385. (22)Igawa, A.; Fukutome, H. B o g . Theor. Phys. 1975,54, 1266.
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The Journal of Physical Chemistv, Vol. 86,No. 7, 1982
calculational method may be used before sections 5 and 6 below. It is also worth emphasizing that these sections, in conjunction with the formal application of Lipkin’s method to chemical bonds, represent the major emphasis of the present research. The real number hk is the expectation value in I-) of the number of particles in state &
hk = (-lak+akl-)
(7)
0 5 hk 5 1, hk = h-k
(8)
and satisfies The values of hk are to be determined by minimizing the BCS vacuum expectation value of the Lipkin Hamiltonian which corresponds to the field Hamiltonian H. The trial form of this state includes the possible choice hk = 1 for lkl 5 lkol = 0 for Ik( > lkol (9) and for this number distribution, the BCS vacuum reduces to the particlehole vacuum of HF theory. The variational principle and stability conditions determine which occupation number distribution best approximates the vacuum state. Thus, as was shown by Valatin’ and Schrieffer,lO the method is a straightforward generalization of the HF method. In the HF method, the form of the number distribution (6) is assumed. In the present paper, the choice of distribution is assumed to be hk = 1 for lkl 5 lkIl
O0 (36)
Note that the summations extend over both positive and (28)Clementi, E.;Veillard, A. Theor. Chim. Acta 1967, 7, 133;Veillard, A. Zbid. 1966, 4,22;Clementi, E.J. Chem. Phys. 1967, 46, 3842. (29) Golobiewski, A.;Hinze, J.; Yurtsever, E. J. Chem. Phys. 1979, 70, 1101. (30)Parr, R. G. ‘The Quantum Theory of Molecular Electronic Structure”; Benjamin: New York, 1964. (31)Hinze, J. J. Chem. Phys. 1973, 59,6424. (32)Bloch, C.;Messiah, A. Nucl. Phys. 1962,39, 95. (33)Zumino, B. J. Math. Phys. 1962, 3, 1055. (34)Baranger, M. Phys. Rev. 1963,130, 1244.
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negative values of k. It follows from the definitions that X ( X , X ? = -x(x’,x)
h(x,x? = h*(x’,x)
(37)
The self-consistent energy v is defined by v(x,x? = E(X,X? + Jdxl dxl’ {V(xtxl;x’,xl’)- V(x,Z1;x1’,x?) h(x,x? (38) and the self-consistent pairing potential p by
The e is the one-body part of the Lipkin Hamiltonian, and is defined later. If e is Hermitean and V is Hermitean and symmetric, then p(x’,x) = -p(x,x?
v(x’,x) = v*(x,x?
(40)
The expectation value (HL) of the Lipkin Hamiltonian can be written in terms of h, x, v, and p as = XJdx dx’([e(x,x? + v(x,x?l h(x’,x) - ~ ( x , x ?x*(x,x?) (41) This last expression may be rewritten in terms of the elements W L )
Vk
=
Vkk
=
tk
+ C(Jkkt - Kkkt)*hk k’
(42)
and pk
= pk,+ = C(k,-klVlk’,-k?xkt k’
fork > 0
(43)
England
formulae can be derived which yield exact values for the X k when the exact $k and hk are known. It is convenient to compute Xk # X1 by using (24) and (25),and to replace the computation of XI with the computation of the chemical potential in the manner of the BCS theory. The expressions for the # X1 are readily derived. The requisite X2 for the case of one bonded pair is expressed as 1 1 1 A, = -E(4) - -E(2) + gE(0) (53) 8 4 where the E(n)are the energies of the projected n-particle states P,J-), i.e. E ( n ) = (-P,,HPnl-) (54) The formulae for two bonded pairs in the present model are expressed as 11 7 19 13 35 hz = --E(@ 96 - z E ( 6 ) + 16E(4) - 12E(2) --E(O) 96
+
1 7 1 3 5 = --E(8) + -E(6) - -E(4) + --E(2) - -E(O) 32 48 4 16 96 1 1 1 1 1 A4 = -E(8) - z E ( 6 ) + -E(4) - -E(2) + -E(O) 384 64 96 384 (55) Equation 53 was derived by Nogami.15 The chemical potential constrains the trace of the density matrix to the desired number of particles. This is expressed as the well-known number equation of BCS theory, namely A3
with
IkAl
tk
= Ckk = Jdx dx’4k*(x) E ( X , X ?
(44)
$k(X?
(kklVlklk{) = I d x dx’dxl dx{ &*(x) &*(x?
X
V(x,x’;xl,xl’) 6kl(x1) &l(x’l) (45) Jkk, = (kk1q k k 9
(46)
Kkkt = (kk1Vlk ‘k)
(47)
The Jkk+and Kkk,are Coulomb and exchange integrals, respectively.30For real orbitals &, /Ik is a sum of exchange integrals. The expression to be minimized is (HL) =
f/ZT{(€k
+ vk)hk + pkxk)
(48)
subject to the supplementary conditions ( 1 - 2hk)2 + (2Xk)’ = 1
(49)
The density variation is carried out for the active fermion states defined by the number distribution (10). For a fixed system $k, one minimizes (HL) with respect to hk and X k , subject to the conditions (49). This gives the expressions h k = ( E k - vk)/Z‘Ek (50) Xk
= -pk/2*Ek
(51)
where E k is the Lagrange multiplier for (49),and is given as the expression Ek
=
(vk2
+ pk2)1/2
2
hk=nINA
(56)
k>lkli
where NA is the number of electrons in active states. The n = NA means that the chemical potential is chosen such that n = N k The explicit relationship between the chemical potential and the number equation is established via the one-body energy of the Lipkin Hamiltonian. According to (26),the generalized HF theory, and the Bogoliubov-Valatin fermion transformati~n,~.~ the one-body energy has the general form = Ik (k(5 lkil O r Ik(> lkAl t k = Ik - b k + (57) lkil < lkl 5 I k ~ l where h k is a renormalization energy and X is the chemical potential. The expressions for Ak and X depend on the particular Lipkin Hamiltonian. The case of one bonded pair, or Nogami theory, is expressed as €k
= 4hk& = A1 + 2&(n + 1) (58) For two bonded pairs, one has Ak = [4& + 12nX3 24(n2 An2 4)X*]hk [2x3 24(n + 6)X4](hk)24- 96X4(hk)3 X = X1 + (2n + 2)X2 [3(n2+ An2) + 6 n - 2]X3+ [4(n3+ 3nAn2 An3) + 12(n2 An2) - 24(n - 1)]X, (59) where
+
+
+
+
+
(52)
Under certain ~ o n d i t i o n sE, ~k ~ is a quasi-particle excitation energy. Generally, it is the excitation energy of a singly exicted state relative to the BCS vacuum. Successive Approximatiom to the Lipkin Hamiltonian. The X k may be regarded as Lagrange multipliers which account for the side conditions (24). In this case, precise
(60) lkAl
An3 = 2
Xk(1 -2hk)
(61)
k>lkII
Expressions for several bonded pairs are derived similarly
Fiekl-Theoretic Methods
The Journal of Physlcal Chemistry, Vol. 86, No. 7, 1982
by using the Bogoliubov-Valatin transformation. For fixed f#)k, hk,and # X,I X is determined to yield t k such that the number equation is satisfied. This gives the expression for the successive approximations to X
Ho = Iuk'ak+a-k
uk(i+l)
= ck(i)
+ k'
(Jkkt
- Kkk,)hk('+l)
(63)
j&(i+l)
= [(uk(i+1))2 + (pk(i+1))2]1/2
(64)
&("+l)
=
(k,-kJVlk',-k?xk~''')
(65)
- ,k(i)) /2.E,cO
(66)
k'
hk(i+l)= xk(i+l)
(jJk(i)
= -pk(i)/2.Ek(i)
(67)
Equation 65 is the gap equation of BCS theory. Given A, we can compute X k # X1 with equations such as (58) or (59). For fixed f#)k, Nogami and Zuckerl' have carried out the calculation in the case of two bound nucleons. The Special Case of One Bond. The cluster wave function that describes one bond is expressed as (2). In this case, the requisite f#Jk and Ck(') may be conveniently determined with a two-configuration MC-SCF calculation. Given (2), we can specify the hk by (6) and the number equation (56):
+ h2 = IC,C2)l/(lCl(2)1+ ICJ2)l) hl = ~C1(2)[/(lCl(')1 IC2(2)1)
(68) (69)
Given the f#)k and h k , we can calculate all quantities required for the field-theoretic methods with the previously quoted formulae.
'/z C
klmn
The
where the notation di)means the ith approximation to x . The quantities involved are computed by use of equations quoted previously
(71)
k
H'= Vk'
1209
(kllVlmn)ak+al+ana,- Cgkhk+ak (72) k
are orbital energies vk' = Ik
+ gk'
(73) and gk' is the HF effective one-electron potential energy
gk'=
lEocc
(74)
( J k i -Kki)
The lEocc constrains the summation to the hole states, i.e., those which occur in the determinant. There are analogous partitions of HL which are induced by the BCS wave function. Gorkov,B Anderson," and Nambue expressed the partition with spinor field operators. The Nambu method is particularly useful for diagrammatic computational methods.'JO Rowe applied the Bogoliubov-Valatin transformation to derive a partition of HL that is useful for equations-of-motion calculations.M Nambu Formalism. Nambu9 discovered a formal method which yields perturbation theory for superconductors in terms of ordinary Feynman diagrams. Spinor field operators are used to express the Lipkin Hamiltonian in a form which admits the ordinary perturbation expansion. His results apply also to the present case. A brief resume is given. Details may be found in Schrieffer.lo If the spinor and its adjoint are expressed as
@k+
= (ak+,a-&
(76)
respectively, then the Lipkin Hamiltonian assumes the form HL = HLo + HL' (77) where
HLo = C
@k+[Vk73
k>O
+ &7l]@k + constant
(78)
5. Field-Theoretic Methods Field-theoretic methods which are based on determinantal wave functions (particle-hole vacua) have proved useful for chemical applications.3M3 The utility arises from the partition of the field Hamiltonian which is induced by the particle-hole vacuum. This is typically expressed as23
H = Ho
+ H'
The
are Pauli spin matrices 71"
(70)
with
T~
(Y '0)
(;
pl,)
and gk =
(35) Kelly, H. P. Adu. Chem. Phys.1969,14,129; Adu. Theor. Phys. 1968,2,75, and references therein. (36) Csanak, G.; Taylor, H. S.; Yaris, R. Adu. At. Mol. Phys. 1971, 7, 287. (37) Lmderberg, J.; Ohm, Y. 'Propagators in Quantum Chemistry"; Academic Press: New York, 1973. (38) McCurdy, C. W.; Rescigno, T. N.; Yeager, D. L.; McKoy, V. 'Methods of Electronic Structure Theory"; Plenum: New York, 1977; Vol. 111, pp 339-86, and references therein. (39) Simons, J. Annu. Rev. Phys.Chem. 1977,223, 15, and references therein. (40)Cederbaum, L. 5.;Domcke, W. Adu. Chem. Phys.1977,36,205, and references therein. (41) Bartlett, R. J.; Purvis, G.D. Phys.Scr. 1980,21, 255, and references therein. (42) Herman, M. F.; Freed, K. F.; Yeager, D. L. Adu. Chem. Phys. 1981, XLVIII, 1. (43) Paldus,. J.:. Cizek Adu. Quantum Chem. 1975.9. 106, and references therein.
73=
c(Jkl Kkl)hl 1
(81)
is the generalization of (74). The unperturbed one-particle Green function is a 2 X 2 matrix whose components are expressed in the interaction representation as Goob = -i(-l~@ko+(t)@kb(0)~l-)
(82) where T i s the Wick time-ordering operator.l0 If t > 0 in all cases
(44)Rowe, D.J. Reu. Mod. Phys. 1968,40,153.
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The Journal of Physical Chemistry, Vol. 86, No. 7, 1982
Go(k,-t) = -i[ i-1 - (hk
+
;)-73
- Xk.711
eXp[i*Bk*t.l] (84)
where 1 is the 2 X 2 unit matrix. It is convenient to impose the following boundary conditions (85)
incorporating the aforementioned changes into the usual r~les.~ For , ~example, ~ the (frequency-dependent) second-order direct self-energy contributions are
.I - -
m
m'
I/
=
z
-'/4
- i ~ , [ G ~ ( m , t ) J fX= ~ +
m, m'eX ICX
Tr [ ( - i ~ , G O ( m ' , t ) ) ( - i ~ , G 0 ( l , - tt=o+73(h'llVlmm') ))] X -
Under these conditions, one obtains the matrix propagator at time zero as (86) Go(k,O) = i ( h k T g + x k 7 1 ) Furthermore, the boundary conditions (85) remove the constant terms in (78) and (79), and also the last operator in (79);lO thus, one works with the terms HLO
=
zo
@k+[Vk73
+ pkTll@k
(87)
m,m'eX lex
hk71l@k
--V4
2 -i7,[G,(m,-t)]t=,+ x m,kX m'cX Tr [ ( - i ~ ~ G ~ ( m ' , t ) ) ( - i ~ ~ G t=O+T,(km' ~ ( l , - t ) ) ]IVlml) X
___----
=
where Tr denotes the trace. If the generalized quasi-particle energies E k are used in the energy denominators, then the perturbation series may be derived upon making two changeslJO from the rules quoted for the HF vacuum.45 Each vertex introduces a factor 73, and each closed loop gives a factor minus one times the trace of the product of the T ~ G ~which ) s form the loop. (The matrices must be kept in the proper order.) For example, second-order linked diagrams for the ground-state energy are
I _/_ _.'I __ ,*-----.t
k
/'=
Tr [ - i ~ , G , ( k , t )i ~ , G ~ ( l , - t ) ] x~ , +
2:
k.k'cX
1jax
TY [ ( - i ~ , G ~ ( k ' , t ) ) ( - i ~' ,~- G t ) ~) ]t=o+ ( l I( kk' lVlll')12/
-
_____
k' =-l/z
+
q k + = Ukak+
- vka-k
ak+ = U k q k +
+ Vk7-k
(94)
where uk = (1 - hk)1/2 vk
= (hk)1'2
u-k = uk v-k = -vk
(95)
The quasi-particle operators defined by this unitary transformation satisfy qk'l-) = 0 (96) In general, the unperturbed Lipkin Hamiltonian can be expressed in the form6,' =
c EsL+L +
(HL)
(97)
S
Eh' + El + Et)
(go)
Tr [(-iT3Go(k,t))(-i~,G,,(1,-t))X
I:
[1/2/1- (h,
Equations-of-Motion Method. The Bogoliubov-Valatin transformation5p6is expressed as
HLO
k
z
- ~~~~]h~(km'IVlrnl)(mliVlh'm')/ ( w + E,' + E, + El) (93)
1,l'oX (Ek f
-
+ E, + E l ) =
m,leX m'eX
(88)
(89)
X
/ 5 J ; ; ; l
l / 2 ) ~ ,
instead of (78) and (79). Note that Tr ( - i ~ ~ G ~ ( k ,=0 )hk ) + h-k
73)
Ern< - w + 6) (92)
-
@k+[gk73-k
(1 +
h ~ ( k ' l l V i m n ~ ' ) ( m m ' i V i k l )+/ ( ~ ,
( m l l V l k ' m ' ) / ( w+ E,' k>O
+EI)=-~/~2
( m m ' l V l k l ) / ( ~+, T , , - w
k,k'cX 1,l'CX
(-i7,G0(k',t))(-h3G0(l',-t ) ) ]t=o+(kk'lVill')(kk'lVll'l)/
where (HL) is defined by (48),and the r s are a unitary transformation of the q's7
ls
= CCsk?ls k
cs+
= CC*sk)ls+ k
(98)
For simplicity, it is assumed here that the q's diagonalize the quasi-particle Hamiltonian, in which case one has2v6g7 HLO
=
WL)
+ cJ%s+%
(99)
S
1, l'eX
( k k ' I v I I ' ~ ) / (+z E~ k r
+ & + Et)
(91)
The notation Tr [-.ItPO+means t > 0 is used to calculate the matrix product, and then t = 0 is used to calculate the trace. The summation limits k E X (k 4 X)mean that k lies in the excited space X ( k does not lie in the excited space X). Dyson's proper self-energy1may be evaluated similarly. In the Nambu formalism, it is a 2 X 2 matrix whose perturbation serieslJOis evaluated with ordinary diagrams by (45) March, N. H.; Young,W. H.; Sampanthar, S. "The Many-Body Problem in Quantum Mechanics";Cambridge University Press: Cambridge, 1967.
The :-: denotes normal ordering2#& with respect to the BCS wave function, and Vklmnis the antisymmetrized twoelectron interaction matrix element2 Vklmn = (kllVlmn) - (kZlVlnm) (101) Given (99) and (loo), we find that the equations-ofmotion are those derived by R o ~ e . Consider, ~ , ~ ~ for example, the quasi-particle random phase approximation.2,44 The excitation operator is defined as
OX+ =
k€X
lex
[Ykl(A)qk+ql+- Z k l ( A ) q l q k l
(102)
The Journal of Physical Chemistry, Vol. 86, No. 7, 7982 1211
Field-Theoretic Methods
and the excitation energies ox are determined from the matrix equation
Wick‘s theorem.2 If HLis expanded in normal order with respect to the BCS vacuum, then H L = (HL) + HL’ C v~,:cz~+u~: -
+ kl
If k,k’ E X and 1,l’ X,then the submatrices A(Hermitean), B(symmetric), and U(Hermitean) are found to be Aklky = b k k d l l r ( E k El) ~ l ~ i J V k l k T l t Ului!Vk,-p,-l,kJ (104)
+
+
Bklk’l’
+
=
UklkV
UIUl’Vkk’,-l,-l’
(105)
= akk’all’
(106)
6. Pseudo-Hartree-Fock Orbital Equations It is straightforward to derive HF-type orbital equations which apparently provide accurate approximations to the rigorous variational orbitals of section 4. The analysis requires the eigenvalue equation (20) and the BCS Ansatz (4). Following Valatin,’ variation of (HL) (48) with respect to &* yields the rigorous orbital equations Kb,x)
d ~ k b )= Chi hCU) 1
(107)
where Kb,x) = ldx’bb,~? h(x’,x) - d . ~ , x ? x*(x’,x)J
(108)
and the hkl are Lagrange multipliers which ensure orthonormality of the &. The density matrices h and x are defined by (34) and (39, respectively, and the energy v and pairing field 1.1 are defined by (38) and (39), respectively. If the relation h l =0 (109) obtains, then Valatin’s commutative case is realized.’ Since v is manifestly Hermitean (see (40)), (107) may be solved in the orbital representation which diagonalizes v. The orbital equations are then pseudo-HF canonical eigenvalue equations V b J )
d d x ) = e k 6kb)
(110)
where and @k are the eigenvalues and eigenfunctions of Y, respectively. The equations are termed pseudo-HF equations because fractionally occupied orbitals are treated. Thus, the one-body energy is renormalized and includes the chemical potential (see (57)), and the two-body energy includes the hk as in (81). Otherwise, (110) may be solved by using an ordinary HF algorithm. The determination of the density matrix and Lipkin Hamiltonian is carried out according to section 4. Upon convergence of this procedure, the field-theoretic methods of section 5 may be applied. There is an additional utility associated with the canonical pseudo-HF equations: the operator v itself induces an ordinary HF-type Green function. Thus, the partitioning techniquesG of perturbation theory may be applied directly to v to simplify the solution of the orbital equations. Such partitioning is the essence of Grimley’s theory of cherni~orption.~~ Although (109) is typically not satisfied, it is reasonable to expect that the @k obtained from (110) are sufficiently accurate approximations to the rigorously determined @k for many purposes. This may be further appreciated by recognizing that it is also possible to realize (110) by using (46)Messiah, A.‘Quantum Mechanics”;Wiley New York, 1964,Vol. 11.
(47)Grimley, T. B. ‘Chemistry and Physics of Solid Surfaces”; CRC Cleveland, 1976;and references therein.
yz c &l(:U-kul: + :uk+u-l+:)(111) kl
where :-: denotes normal ordering and (HL) and HL’ are defined by (48) and (loo), respectively. The matrix element vkl
= l d x dx’&*(x) v ( ~ @i(x? 3
(112)
occurs in the number-conserving part of the one-body Hamiltonian, while the matrix element ~ k =i
Jdx dx’@k*b)d x , x ? &(x?
(113)
appears in the number-nonconservingpart. The canonical pseudo-HF equations are therefore equivalent to diagonalization of the number-conserving part of the one-body field Hamiltonian. Since particle nonconservation in BCS theory is only essential to the definition of a convenient vacuum, the utility of (110) appears promising. 7. Conclusion The BCS wave functions described herein are convenient alternative representations of restricted cluster-type wave functions. For cases of interest, they may be computed with proven methods: even-replacement MC-SCF orbital algorithms; Hartree-Bogoliubov density matrix algorithms; and Nogami’s successive approximation method for the Lipkin Hamiltonian. It is believed here that the generalized valence bond (GVB) method4s could be used to determine the requisite orbitals and density matrices, at least to a first approximation. Quantum chemical field-theoretic methods are at least competitive with the configuration interaction (CI) method.42*4s51 It is possible to predict some success for BCS-based methods in this context. First, consider that HF-based methods work well when the spectrum is nondegenerate and no bonds are broken or formed. Given the multiplicative quasi-separabilitf2 of the BCS wave function, and the isomorphism of the BCS and HF field theories, the BCS methods are expected to work equally well when bonds are broken or formed. This proper description of bonding is also characteristic of the generalized valence bond method.ls Roughly speaking, BCS-based methods can be equated to GVB-based methods, although, owing to its cluster type, the BCS wave function is more flexible than the corresponding GVB wave function. (Although it is not the subject of this paper, a BCS-based CI method could be developed. The cluster-type wave function would serve as reference, and would be computed self-consistently.) Thus, the BCS field-theoretic methods should be at least competitive with GVB-based CI methods. Description of degenerate spectra requires model spaces and/or flexible methods. Complete model spaces appear in first-order type CI methods,53and in several diagrammatic many-body perturbation theories.54-60 Arbitrary (48)Bobrowicz, F. W.; Goddard, W. A. “Modern Theoretical Chemistry”; Plenum: New York, 1977;Vol. 1x1, pp 79-127. (49)Bartlett, R. J.; Silver, D. M. “Quantum Science”; Plenum: New York, 1976;pp 393-408. (50)Wilson, S.; Silver, D. M. Phys. Rev. A 1976, 14, 1949. (51)Bartlett, R. J.; Shavitt, I. Chem. Phys. Lett. 1977, 50, 190. (52)Primas, H.‘Modern Quantum Chemistry”;Academic Press: New York, 1965;Vol. XI, pp 45-74. (53)Schaefer, H. A.;Clemm, R. A.; Harris, F. E. Phys. Rev. 1969,181, 137..
(54)Brandow, B. H.Reu. Mod. Phys. 1967,39, 771;Adu. Quantum Chem. 1977, IO, 187. (55)Sandars, P.G. H. Adu. Chem. Phys. 1969,14, 365.
J. Phys. Chem. 1982, 86, 1212-1213
1212
model spaces are no particular problem for typical CI methods,61and have also recently been incorporated into diagrammatic perturbation theoryF2 Multiconfigurational equations-of-motion theories exist as well.42 The BCSbased method will work well only when the degenerate states are approximately described by a BCS Ansatz. Thus, at best, only special open shells could be considered. The pseudo-HF method can be precisely defined for any BCS wave function, and offers BCS-based orbital equations which are effectively HF equations. In short, the BCS fractional orbital occupancies appear explicitly in the electrostatic energy of a single HF-type Fock operator whose one-body energy is renormalized. Given the success and relative simplicity of quantum chemical HF orbital algorithms, the pseudo-HF method holds promise. This is of practical and aesthetical interest: if the pseudo-HF method proves equivalent to the general BCS orbital equations, then a BCS method is available whose orbital and field theories are isomorphic to HF theories. (56) Oberlechner, G.; Owano-N-Guema; Richert, J. Nuouo Cim. B 1970, 68,23. (57) Johnson, M. B.; Baranger, M. Ann. Phys. (N.Y) 1971, 62, 172. (58) Kuo, T. T. S.; Lee, S. Y.; Ratcliff, K. F. Nucl. Phys. A 1971,176, 65. (59) Lindgren, I. J.Phys. B 1974, 7, 2441. (60) Kvasnicka, V. Ado. Chem. Phys. 1977, 36, 345, and references
therein. . ~.~ ~
~~~
.
(61) Shavitt,I. “Modern Theoretical Chemistry”;Plenum: New York, 1977; Vol. 111, and references therein. (62) Hose, G.; Kaldor, V. J. Phys. B 1979,12, 3827; Phys. Scr. 1980, 21, 357.
The ab initio HF and GVB methods have been successfully applied to large systems which represent real surfaces. The HF calculations have been largely concerned with photoioni~ation,~~ whereas the GVB studies have pursued catalytic reaction mechanisms.64 The applicability of the rigorous BCS method to such systems should be at least comparable to that of the GVB method. The infinite crystal%and pseudopotential% techniques developed there will also work with BCS wave functions. If the pseudo-HF method proves useful, then the BCS method could be applied to large systems almost as easily as a corresponding HF method, and would, in principle, provide the simplest possible ab initio method for the investigation of reactive processes. Any of the model potential^^^ or p a r t i t i ~ n i n g developed s ~ ~ ~ ~ for single Fock operators could be used. If desired, semiempirical parameterizations could be introduced straightforwardly.
Acknowledgment. Acknowledgment is made to Dr. David Silver for helpful comments. This work was supported by the Graduate School, University of Wisconsin-Milwaukee. (63) Bagus, P. S.; Hermann, K.; Seel, M. J. Vac. Sci. Technol. 1980, 18, 435. (64) Upton, T. H.; Goddard, W. A. ‘Chemistry and Physics of Solid Surfaces”; CRC: Cleveland, 1981; and references therein. (65) Upton, T. H.; Goddard, W. A. Phys. Reo. E 1980,22, 1534.
(66)Melius, C. F.; Olafson, B. D.; Goddard, W. A. Chem. Phys. Lett.
1974,28,457. (67) Suritalski, J. D.; Schwartz, M. E. J. Chem. Phys. 1975,62,1521; 1976,64, 4245.
Why Does the Diatomtcs-in-Molecules Method Appear To Fail for H,? Philip J. Kuntz* and Christian C. Chang Hahn-h<ner-Institut fiir Kernforschung, Berlin QmbM Bereich Strahlenchemie, P 1000 Berlin 39, Federal Republic of Germany (Received: October 27, 1981)
Using extended-basis valence bond calculations, we computed the projections of the X ‘Zs+ and b 3Zu+ states of H2 onto several diatomics-in-molecules(DIM) asymptotic basis sets as a function of internuclear distance. From these it is concluded that 2Pstates of H must be included in the DIM basis in order to represent the triplet state adequately at short distances. This suggests that previous DIM calculationson H4may be inadequate and that new work with a larger basis is necessary.
The method of diatomics-in-molecules (DIM)1-3 is fast becoming a very useful tool for representing potential energy surfaces in chemical dynamics ~ t u d i e s . ~It, ~has enjoyed considerable success with many systems, especially H3+;596however, some molecules present d i f f i ~ u l t i e s . In ~~ this paper, we argue that a possible explanation as to why the DIM energy of the coplanar H4 molecule in its ground electronic state is much too low compared to ab initio is that the previous DIM calculations may not (1) F. 0. Ellison, J.Am. Chern. SOC., 85, 3540 (1963). (2) E. Shiner, P. R. Certain, and P. J. Kuntz, J. Chem. Phys., 59,47 (1973). (3) M. B. Faist and J. T. Muckerman, J. Chem. Phys., 71, 225 (1979). (4) J. C. Tully, Ado. Chem. Phys., 42, 63 (1980). (5) J. C. Tully, Ber. Bunsenges. Phys. Chem., 77, 557 (1973). (6) C. W. Bauschlicher,Jr., S.V. O’Neil, R. K. Preston, H. F. Schaefer, 111, and C. F. Bender, J. Chem. Phys., 59 1286 (1973). (7) M. B. Faiat and J. T. Muckerman, J. Chem. Phys., 71,233 (1979). (8) C. W. Eaker and C. A. Parr, J. Chem. Phys., 65, 5155 (1976). (9) C. W. Eaker and L. R. Allard, J. Chern. Phys., 74, 1821-3 (1981). 0022-3654/82/2086-12 12$0 1.25/0
have used the correct basis. Our analysis serves as an example of how one can go about obtaining self-consistent DIM models for other systems as well. The structure of a DIM model depends on the number and type of polyatomic basis functions contained in it; hence, the numerical results of even an empirical DIM model may be sensitive to the choice of basis. Until recently, there has been no prescription for selecting an appropriate basis set. This is partly because there is as yet no variation principle for the method, so that the errors associated with a particular model are difficult to estimate. Furthermore, the DIM method is often applied semiempirically in the hope that the use of exact diatomic potential curves will compensate for other shortcomings of the method. In other words, the basis set is too often viewed as a convenient framework on which to hang the DIM model but not as something important in itself; afterall, the DIM wave functions are hardly ever used. This viewpoint is, however, fundamentally wrong and 0 1982 American Chemical Society
