2488
B. A. MATSENAND A, A. CANTU
Spin-Free Quantum Chemistry.
VII.
The Slater Determinant1
by F. A. Matsen Molecular Physics Group, The University of Texas, Austin, Texas
and A. A. Cantu Department of Chemistry, University of Alberta, Edmonton, Alberta, Canada
(Received June 11, 1968)
There exist spin-free analogs for Slater determinants including those with different orbitals for different spins. The spin-free analog of the resolution of a Slater determinant into pure spin states is the resolution of its spinfree analog into pure permutation states. Like a Slater determinant, its spin-free analog can accept an assignment of no more than two electrons to a spin-freeorbital. The spin-free analog of a Slater bond function which lies in a pure spin state is a spin-free bond function which lies in a pure permutation state. Just as for a Slater determinant, the expectation value of a spin-free operator over its spin-free analog is the sum of contributions from pure states with “Sanibel” coefficients. Just as for a Slater determinant, its spin-free analog is a basis for a variety of self-consistentfield calculations.
I. Introduction The earliest treatments of the N-electron problem were based on the theory of the symmetric group.za I n 1929 Slater2bproposed his determinantal method for the N-electron problem, a method which did not require a knowledge of group theory and which was inherently simple. The Slater determinant has been the basis of most of the development in quantum chemistry during the past 40 years. However, the Slater determinant method requires explicit use of spin even for systems for which a spinfree Hamiltonian is sufficient. The close association of the N-electron problem with spin required by the Slater determinant method has led to the development of a spin-oriented language. The use of this spinoriented language has led many chemists to believe that spin plays a dynamical role in the greater part of quantum chemistry. The present series of papers3 has for its purpose the establishment of a viable spin-free quantum chemistry. Such a formulation has an interest and utility of its own, but, in addition, its existence should have the effect of demoting spin to its proper place in quantum chemistry. Since most of the theoretical development in quantum chemistry has been in terms of Slater determinants, we exhibit a apin-free analog of the Slater determinant. This exhibition occurs in section IV. Sections I1 and I11 contain the required group theory. I n section V are exhibited the spin-free analogs of a number of selfconsistent field wave functions constructed on Slater determinants.
11. Primitive Kets and Their Invariants Let the vector space V ( 7 )which is spanned by a basis ~ ( 7 )~ : ( 7 =) {ly;i),i
ranging)
be an invariant vector space with respect to S, The Journal of Physical Ch.emistry
(2.1)
the
group of permutations on N objects. Furthermore, let this space be generated from a given primitive lret 17) by applying permutations of S N to the electron coordinates of 17). For this discussion we take 17) to be an orbital product of independent orbitals
.4K*. .4J
(2.2) Here the orbital in the first position is assigned to the first electron, etc. The orbital product ket is characterized by the partition4 17) =
141,
where y k is the number of & in
Note that
N
=
&k
17).
k
We arrange the orbitals in
1 y ) such that
7 ’ 17 2 2
*
.
I
2
(2.5)
The partition { y f is called the invariance of 17). The invariance group of 17) is G Y
=
p l y , .
= SY,@&@
. .,GYn(Y)) . . @,ST, 1
c s,
(2 * 6)
where XY,is the group of permutations on the 7k electrons assigned to c $ ~ and where (1) This research is supported by the Robert A. Welch Foundation of Houston, Texas. (2) (a) See, for example, €1. Weyl, “Group Theory and Quantum Mechanics,” 1st ed, 1928. Reprinted by Dover, New York, N. Y. (b) J. C. Slater, Phys. Rev.,34, 1293 (1929). (3) The spin-free quantum chemistry series consists of the following papers: paper I : F. A. Matsen, “Advances in Quantum Chemistry,” Vol. I, P. 0. Lowdin, Ed., Academic Press, New York, N. Y., 1964. Paper 11: F. A. Matsen, J . Phys. Chem., 68, 3282 (1964). Paper 111: F. A. Matsen, A. A. Cantu, and R. D. Poshusta, ibid., 70, 1558 (1966). Paper I V : F. A. Matsen, ibid., 70, 1568 (1966). Paper V : F. A. Matsen and A. A . Cantu, ibid., 72, 21 (1968). Paper VI: F. A. Matsen and D. J. Klein, ibid., 73, 2477 (1969). Paper VIII: F. A. Matsen and M.L. Ellsey, ibid., 73,2495 (1969). (4) A partition of N is a set of integers whose sum is N .
SPIN-FREE QUANTURI
Gar\d = The order of
2489
CHEMISTRY
14,GUYE GY
(2.7)
firl[Al
1 =j
GYis nlYl
=
-y’!y2!,, . $!
(2.8)
See Table I.
-
N!
such that
According to (2.18), if V(ya; [A]). Further
The vector space generated from the orbital product ket defined in (2.2) is (2.11)
where (2.12)
and P t y is one of the left coset generators for S , with respect to GY. The dimension of V ( 7 )is equal to the index of GY
o if { r )> [ A ]
{y}
j [ r ) [ A= l
(2.10)
= PiYj7)
XP [AI
(2-17)
for N = 4
f(rl[A1 =
(2.9)
V(y): B ( y ) = {ly;i) i = 1 t o p ) ]
(’11
hlurnaghan6 has shown that6
We define the idempotent operator
14
nPXP
is the number of times occurs in Here xplYJand xP[’] are the characters of the pth class in rlY1 and I?[’], respectively, and n,, is the number of elements in the pth class. See Table 11.
Table 11:
EYIY) =
c
(2.18)
> [XI, V ( 7 )does not contain
I if {y] = [ A ]
(2.19)
and = f[’] if { y f = ( I N }
(2.20)
The Pauli-allowed permutation states are characterized by partitions of the form [ A ] = [2”, l N P 2 p ] . Thus by (2.18) for { y 5 [X 1, the primitive ket can only have invariances { y } = { 2 O , lNm2*} with q 5 p. This implies that 17) is restricted to having q 5 p doubly occupied orbitals occuring among the first 2‘ electron coordinates. This is the spin-free exclusion principle which states that no more than two electrons can be assigned to a spin-free orbital. The frequencies’ of (2.17) for the allowed permutation states are given by
The vector space V(y) can be decomposed into a direct sum of minimal invariant subspaces (permutation states) under S ,
By (2.20) and (2.21) fllNlhl = j h l =
where V(y.;[Xl):
p!(N - p B(rfl;[AI) =
{ / r u ; [ A ] r ) r,
(2.15) = 1 to jrA1/
+ 1) + l)!
N ! ( N - 2p
(2.22)
There follow four important tool theorems. Theorem I
and [ A ] is a partition of N . The decomposition of V(y) induces a reduction of I’lrl, the representation of X, on V ( y ) . Thus rlrl =
flrl[hIr[Al [A 1
(2.16)
where is an irreducible representation of SN and the frequency
( 5 ) F. D. Murnaghan, “The Theory of Group Representation?’ John Hopkins University, Baltimore, 1938. (6) ( 7 ) > [ A ] means y‘> A 1 or if y1 = A 1 then y2 > X2, etc. (7) J. C. Schug, T . H. Brown, and M. Karplus, J. Chem. Phys., 35, 1873 (1981). See also F. A. Matsen, Molecular Physics Group Technical Report, Austin, Texas, 1962, eq 871.
Volume 79, Number 8 August 1960
2490
P. A. MATSENAND A. A. CANTU
By eq 2.9
The proof follows that of theorem I1 making using of theorem 111. Murnaghan (ref 5, p 151 ff) has shown that for { p f =
(N-wj =
flfil[Al
where n p i r lis the number of elements in the intersection of the pth class and G”. According to Murnaghan (ref 5 , p 94) (2.25)
- {
(This is also shown in the Appendix of paper VI of ref 3). Note that for the nonzero frequencies in (2.34), [A] must be of the form [2., 1 N - 2 p ] with p 5 p .
Theorem V 11” =
We substitute (2.25) into (2.24) and use (2.13) to obtain X[y&Y)
1
npXpn(~lXp[Al
=-
f
Theorem I I . There exist irreducible representations of SN such that [Eyl,s[X1
{
=
a,, for r,s
=
o for r,s > j(”)[’l
= 1 to j(’1 [’I
(2.26)
Since E” is idempotent, we can choose a representation of &’ such that
a,, for T,S
{I- 0 for
[&”I,,[’l
T,S
m
= 1 to
>m
ell[’]
[AI 5 131
for ( p ] = {2”,1N-2r] (2.35)
Since 1” is an element in the group algebra we can expand it in the matric basis (section 111)
= (’1 [XI
N!
1 for [A] I {,E] = {2fit1N-2~] 0 otherwise (2.34)
(2.27)
11” =
C C Cs 1x1
(2-36)
T
We substitute (2.33) and (2.34) into (2.36) to obtain the theorem. We shall see in section IV that the group G” is to be associated with a spin ltet of invariance ( N - p , p } . 111. The Decomposition of V (y) For the decomposition of V(y), we employ the matric bases of the permutation algebra which contains elements of the form
The representation defined by (2.27) is said to be canonical to By (2.27) X[Xl(&“) =
[&”],,[hl
= rn
(2.28)
T
By Theorem I =
(2.29)
f(rl[h
(2.29) together with (2.27) constitutes the theorem. For an invariance group G’ of a ltet Ip) we define an antisymmetric idempotent
1
7” = __ ni’)
c
e
(PU”)PU”
where [Pa-l]s,[xl is the s,rth element in the f i x ] x f[’] matrix which represents Pa-I in I’[’]- rIxl is chosen canonical to &’ (see theorem 11, section 11),i.e. e,,[’]&Y
=
j(’l[’l
(3.2) (3.3)
The basis elements in (2.15) are defined by
(2.30)
17s; [A IT)
= ers[A1/
(3.4)
Then by (2.10), (3.2), (3.3), and (3.4)
Theorem III x[Al(q~)
i=.
f(PI[);]
17s; [A
(2.31)
XIAl(llr) =
1
nlr’)
c
e
(2.32) The proof then follows that of theorem 11. TheoTem I V . There exists an irreducible representation for X, such that
i
~ r s ‘ ~ ]
=
a,, for r,s
= 1 to f
o for r,s > j 1”) [‘I
The Journal of Phgsical Chemistry
=
7)
1Is
=o
If
(3.5)
> jiYl[’l
(3.6)
It follows from (3.4) and the properties of unitary matric bases of the permutation algebra
(Pa?X[’w?
a
=
I79
= e,,[’]jy)
where I?[’] is the representation conjugate to I?[’’. By (2.30)
[II”
{
(filfi]
(2.33)
( y s ; [ ~ ] r / ~ l y s[ ’~;’ l r ’ )= (y[e,,[AIHe,~,~[A’l~y) f[Al
=
S(b1,
[h’I)qTlT’)
E
c [~u-lls’SIA1(yl~PuJy) a
s’,s = 1 tof(Y)[AJ (3 ’ 7)
By (3.7) the representation of H in the matric basis of V(y) is factored into blocks, each characterized by [h]. The [Alth block is factored into f[’] identical blocks
2491
SPIN-FREE QUANTUM CHEMISTRY each characterized by r and each of dimension X fiY1[’l. The eigenkets are of the form
(r;KbIr) =
Irs;[Xlr)(rslK[hI)
f‘yJ[hl
(3.8)
with eigenvalues E ( y ; K [ X ] ) . The index r = 1 to f i x ] labels the degenerate eigenkets. Because of the degeneracy it is sufficient to consider the subspace V(r;[X]l):Ilys;[X]l), s = 1 t o j ‘ y
lyl;[Xl1) = en[”/y) E V(y;[X]l)
Ir.;[Xl)
(3.10)
= u%)
where ,l
=
Y
c
I [’I us
(3.11)
e,,[’
1
5-
is called the sigma projector. The modulus of a minimal invariant subalgebra
AIx1 of the algebra of the symmetry group A ( S N ) is defined by
(3.19)
By (2.14) and (2.18) irrespective of the representation chosen for the construction of ell[’] Irl;[hIQ = 0 if { 71
(3.9)
The fact that V(y; [All) is not a minimal invariant subspace is of no consequence here. A general vector in V(y; [All) is
,[XI
where
9
> [XI
(3.20)
Consequently
Note that if { y ] = f p ] then (3.21) has the simple form of (3.17) and is in the pure permutation state [A] = [y]. If we now relax therestriction of the double occupancy in 17) to { y } < { p ] and wish to retain the simple form of (3.17) we do this at the expense of projecting an impure permutation state. This in fact is the nature of the unrestricted Hartree-Fock scheme. (See Table IV.) If one projects on l y p ) with e[h1, the projected ket lies in a pure permutation state. This corresponds to the projected unrestricted Hartree-Fock and the extended Hartree-Fock schemes. (See Table IV.)
1V. The Spin-Free Analog of a Slater Determinant A Slater determinant is defined by (3 * 12) The identity element is a sum of the moduli in the several subalgebras. Thus g =
(3.13)
p 1 [’
where
I
It follows that a primitive ket can be decomposed into a sum of components lying in the several permutation states. Thus
(’,) is an eigenket to 8,with an eigenvalue of
N
M = - - p
2
17) = 47) =
c Ir;[U
(4.3)
From the theory of determinants
(3.14)
1’ 1
where
only if
Ir;[Xl)= e[’llY) =
0 iffIYl[’l
=
0
(3.15)
ly;[X]) is called an immanant.* The projection of the antisymmetric projector (2.30) into A[’] is by (2.35) ,f[Xl
=e -
v ~r= ell[’] for [XI =
o for
[XI
I {?I
1”-q
0 5 (1 5 p (4.5) That is, no more than two electrons may be assigned to the same orbital. The maximum value of 8 for ( y j = { P , 1N-2fj is { y ] = {2f,
(3.16)
K = -N- q
> {p}
2
(z)
The ket
Irr) = l/n?;T?”IT)
[ y) has an invariance
(3.17)
is shown in section IV to be the spin-free analog of the Slater determinant. By (2.35) and (3.4), (3.17) becomes
The Slater determinant is not an eigenfunction of I n fact its resolution into pure kets is given by
$,
(8) R.D.Poshusts and R.W. Kramling, Phys. Rev., 167, 139 (1968). Volume 78,Number 8 August 1060
E‘. A. MATWEN AND A. A. CANTU
2492
where G” is the invariance group of ( N - p , p } . Then
Table 111: The Decomposition of Slater Determinants and Their Spin-Free Analogs [ y p ) for N = 4
(z)
(7)
M
I7)
Id
1x1
((:>lGlc@>
8
=
c
e
lp)
(mYIHP:lr)
with
(p] =
(4.15)
It follows from (4.7) that
where
Here
I CZ>N= os1(5,)
(4.8)
and where uSMare the “Sanibel” coefficients. Sasaki and Ohno’b have shown that for orthonormal orbitals and I T 1 =
bNl
where Os is the LowdinQspin projection operator w BM
A Slater bond function is a linear combination of Slater determinants and is an eigenfunction of k2. It provides a quantum mechanical model for a single molecular structure, D,. For a molecule with p bonds, X = ( N / 2 ) - p . For a singlet state, D, contain p = N / 2 bonds and the bond function is given bylo-12 (4 lo) 1
(4.17)
Theorem V I . The spin-free analog of a Slater determinant (;)) is as given in (3.17)
I
(4,18)
1
h@ITd
where R” permutes pairs of orbitals of opposite spin that are connected by arrows in D,. The number of independent singlet bond functions which can be formed from I(:) with ( y ) = f l N )is
((3;.lHlO
;.‘> = c (p,)xxMPu174 (4.12)
=
n(”Y7lr”Hr”lr)
= n%lHv”/y) =
c
E
(Pu,(ylWIy)
(4.19)
U
which is in agreement with (4.15). We now proceed to show some of the properties of yp) which parallel those already established for its spin analog (Z)). The purpose of this is to show that these properties arise merely from properties of the permutation algebra and not from spin. By (3.21)
1
The expectation value of a spin-free Hamiltonian over a pair of bond functions is
I
The equivalence of (i)) and y p ) is established by showing that they give the same matrix element for a spin-free Hamiltonian. Thus
1
a
Schemes for the computation of the Pauling numbers (Pa) have been proposed. The expectation value of a spin-free Hamiltonian over a Slater determinant is
c
((Z)IHI(’r>> =
E
(P,)(ylHPuIy)(CIlPaII*)
(4.13)
Now 1 for Pa c G” 0 for Pa r G” The Journal of Physical Chemistry
(4.14)
(9) P. 0.Lowdin, Phys. Rev., 97,1509 (1935). (10) J. C.Slater, ibid., 38, 1109 (1931). (11) A. Sherman and H. Eyring, J. Amer. Chem. floc., 54, 239 (1932). (12) L. Pauling, J. Chem. Phys., 1,280 (1933). (13) L. Pauling and E. B. Wilson, “Introduction to Quantum Mechanics,” McGraw-Hill Book Co., Inc., New York, N. Y.,1935. (14) H. Eyring, J. Walter, and G. E. Kimball, “Quantum Chemistry,” John Wiley and Sons, Inc., New York, N. Y., 1944. (16) F.Sasaki and K. Ohno, J . Math. Phys., 4,1140 (1963).
SPIN-FREE QUANTUMCHEMISTRY
2493
Table IV : Spin-Free Function for Self-consistent Calculation" Primitive ket invariance
1. 2. 3. 4. 5.
6.
7. 8. 9. 10.
a
Hartree Restricted Hartree-Fock Unrestricted Hartree-Fock (spin polarized) Projected unrestricted Hartree-Fock (orbitals determined from 3 before projection) Projected extended Hartree-Fock (orbitals determined after projection) Optimum projected unrestricted HF (orbitals from 3, u projector optimized) Optimum projected extended H F (orbitals from 5, u projector optimized) Goddardlg GI method* Poshusta-Kramlings immanant Poshusta and Kramlings optimum extended unrestricted H F (orbitals and u projector are simultaneously optimized)
Projeoted ket characterization
Projector
[2N/2] U [3,2N/2-2,1] u. .. [N] [2Nq [lN] u [ 2 , 1 N 4 ] u . . . u [2N/2]
u
{2 N q
/;yJ {IN\ {IN]
{IN)
I YJ { YI I Y)
The u projector is given by (3.11)and can also be expressed in terms of the structure basis3 Y 1 1x1
.[XI
Y
.sel,Ixl =
= 8
I [AI
cap K
The irreducible representations used in this matric basis is the Young orthogonal representation.
We shall see that the spin-free analog of the resolution of a Slater determinant into pure spin states, eq 4.7, is the resolution of its spin-free analog into pure permutation states, eq 4.20. See Table 111. For
{ p } = {2'",lN-"'}0 5 and [XI
p
5 N/2
(4.21)
5 [p] it follows that
= 2/,(')n("l 7'j&"y)
[XI = [2p,lN--2p] 05p
5
p
5 N/2
(4.22)
For [XI = [2p, 1N-2p]and { y ] 5 [XI it follows that the invariance of Iy) is restricted to
{ y)
Consequently (4.20) is the spin-free analog of (4.7). The spin-free analog of a singlet Slater bond function is, using (4.18)
=
{ 29,1N-24)0 5
q
5
5
p
p
5
N/2
(4.23)
Note that (4.23) contains the spin-free exclusion principle which excludes assignments of more than two electrons to a single spin-free orbital. This is in agreement with eq 4.5. Only the permutation states specified by q 5 p 5 p are contained in yp). I n (4.20) y l ; [ A ] 1) is the spinfree analog of y S) where
1 I(#);
I
N X = - - p 2
(4.24)
The highest [XI is given by [2',1N-2M]. Correspondingly, the lowest value of X is
N
M = - - - p
2
(4.27)
where { p ] = {N/2, N / 2 ) and n('l = (N/2!)*. Furthermore, I&"y) = &"ly) where E" is the symmetric idempotent of the group G" which exchanges the electrons in each bond in D,. Here {xf = { 2 q = { 2 q
(4.28)
n("I = 2N/2 = 2P
(4.29)
/ p ) = apapap,we
have { p } = { 3,3 f ,
and For example, for = 36 and
nlPl
1 36
7" = -(B
- (13) - (15) - (35) + (135) +
(153))(9
- (24) - (26) - (46) + (246) + (264))
A possible structure is
(4.25)
The lowest [XI is given by (2P,1N--2f). Correspondingly, the highest value of S is
K = -N- q 2
(4.26) Volume 73,Number 8 August 1960
2494
F. A. MATSENAND A. A. CANTU
For { y ] = { l N )the invariance of I&*y) = { 2 N / 21. Then by (4.20) and (4.27)
&“IT)
is
On thesubstitutionofp = ( N / 2 ) - S a n d p = ( N / 2 ) M , we obtain the Sasaki-Ohno formula (4.17).
I
We have thus shown by formal proof that yp) is the and have spin-free analog of the Slater determinant are demonstrated that properties obtained from easily established using yfi) indicating that spin is not a dynamical factor in the validity of these properties.
(z)
We see that / E X y[All) ; lies in a pure permutation state with [ A ] = [ 2 N l 2 ] ; it is the spin-free analog of a Slater bond function with S = 0 (4.10). The independent spin-free bond functions span the space V(y;[A]l) (3.9). The dimension of this space is by (2.22)
in agreement with (4.11). We note that
N”P”lY)
Jwx) =
(4.32)
where
N” =
n m?I
(4.33)
px =
n(”J&”
(4.34)
and
are the antisymmetric and symmetric Young operators for the tableau T,[xl. (4.32) is, to a phase factor, the definition of a spin-free structure operator employed in papers I and 111. I n paper I11 it was shown that (YP;xlHlyP;x’) =
c (Pa)”*4YlH~alY) (4.35)
with the same coefficients as in (4.12). To derive the Sasaki-Ohno formula, eq 4.17, we note that by (4.20), (3.19),and (3.20)
1
(z)
V. Self-Consistent Wave Functions The self-consistent field procedure is a device for obtaining optimum orbitals for description of atoms and molecules. In the conventional procedure (HartreeFock) a wave function is constructed from one or more Slater determinants. Poshusta and Kramling have given a spin-free formulation of this procedure. I n Table IV we identify the spin-free analogs of several wave functions which are taken as bases in the computation of self-consistent field orbitals. Functions 1 and 3 do not lie in pure permutation states so their physical significance is open to question. Function 3 has been used in the computation of spin density.lB Kets with { y ] = ( 2 N / 2 f for which only the state [A] = ( 2 N / 2 )(singlet) is contained are called closed shell kets. Kets with { y } < { 2 N / 2 }are called open she12 kets. Kets with { y ] = { l Nf are called di$erent orbitals for different spin kets. In this table functions 1 through 7 are those commonly used to describe the “singlet” ground state of a system. (Here “singlet” has no meaning for functions 1 and 3). Functions 8 through 10, as formulated here, apply to the description of any permutation state. To adapt the first seven functions so that they describe “excited” states one then has to consider different kets I y) and appropriate projectors, In common practice the 17) considered for the “excited” states are formed by promoting electrons into “virtual” orbitals forming what we called open-shell kets. Of the kets in the list, (10) has the greatest variability and so should yield the best energy.
Summary I n this series we have employed the symmetric group to develop a spin-free quantum chemistry. I n this paper we have used the symmetric group to obtain a spin-free analog of the Slater determinant. The symmetric group has also been used as a tool in the conventional spin forrnulati~n.’~+~
where
and (4.38)
For { y ] = { l N ) and orthonormal orbitals, (4.38) becomes
(N W[hl(rJ
=
- p ) ! p ! ( N - 2 p + 1) p!(N - p
The Journal of Physical Chemistry
+ l)!
(4 * 39)
(16) A discussion of this point together with references is given in paper V. (17) M. Kotani, A. Ameniya, E. Ishiguro, and T. Kimura, “Table of Molecualr Integrals,” Maruzen, Tokyo, 1955. (18) M. Kotani, “Handbuch der Phyaik,” VoI. 37, Part 11, Springer, Berlin, 1961. (19) W. A. Goddard 111, Phys. Rev., 157, 73,81,93 (1967). (20) F.E. Harris, “Advances in Quantum Chemistry,” Vol. 111, P. 0. Lowdin, Ed., Academic Presa, New York, N. Y., 1967. (21) J. J. Sullivan, J. Math. Phys., 9,1369 (1967). (22) G. A. Gallup, J. Chem. Phys., 48,1752 (1968).
