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Wave function constructed from complex hybrids
Volume
3. number
5
WAVE
CWMICAL
FUNCTION
PHYSICS
CONSTRUCTED E. BR&DAS
Quantum Chemistry
Group. Received
LETTERS
FROM
May 1969
COMPLEX
HYBRIDS
and 0. MARTENSSON Uppsakz University. 19 February
Uppsaka. Sweden
1969
The influence on the total wave function of an atomic basis consisting of complex hybrids is cousidered. The complex character of the wave function in the MO-LCAO scheme ss well as in the VI3 method is derived, and its consequences with respect to time reversal symmetry are discussed.
1. INTRODUCTION Hybrids formed by linear combination of atomic orbitals of s- and p-type are very useful at least for a qualitative understanding and explanation of the chemical bond and other concepts related to this, such as bond direction and bond angle. The bond is described by so-called perfect pairing of two electrons, one from each of two hybrids creating the bond. Conventional hybridization, however, has the limitation that the valence angle definition, which is based on the orthogonality between the hybrids, does not permit vr-lence angles less than 90°. Moreover, equivalency as regards contents of s-orbital exists only for a true tetreonal configuration. A redefinition of the bond (valence) directi= concept and the permission of complex elements in the hybridization matrix (which we consider unitary unless otherwise stated) lead to a procedure, complex hybridization [l], which removes several of the difficulties connected with conventional hybridization, but introduces other problems and aspects which are to be discussed. Using +&e notation of ref. [l], the hybrid of center ,u, numbered k, will be expressed by h;
where 4;
=
ztlah C#J;;
g = 1,2 ,...,
is an atomic orbital of center p, for instance described
N, by
(2)
andAC’={aklisth e unita.y hybridization matrix of the same center, q finally depending on the special choice of ybrklizafion (for orbitals of s- and p-type 4 = 4). The elements a - may be complex and are obtained such that they fulfill given-bond-angle conditions for the atom (ten$ er) & From this it follows that, when complex coefficients occur in the hybrid, these can not in general he factored out, merely forming a phase factor of an otherwise real hybrid. One can, however, without restriction in the first column of AN will be real and positive. choose the phase so, that the element {a[l} A total trial wave function of a molecule may be constructed from the hybrids (1) =h different ways. We can form molecular orbitals by linear combinations of the hybrids, and thereafter take the determinant of the product of MO’s. More natural in this context is to form valence bond gemin&s from the appropriate hybrids and then antisymmetri:ce the whole geminal product. In the following we wiU commat on some aspects on the wave function built up from complexhybrids.
315
Volume
2. THE
3, number
MO-LCAO
5
CHEMICAL
PHYSICS
LETTERS
May 2969
METHOD
In the molecular-orbital method we consider the complex hybrids {I#) in (1) as the basis functions for the total wave function of the system considered. The molecular orbitals are then given by N
qlJ
N
C qp =fl;
j=l,2
,...,
(3)
n,
/l=l
where N is the number of centers, n is the number of valence electrons, and qp indicates the choice of hybridization on atom p_ From (1) we obtain ~j expressed in the atomic orbitak $B. *j
=
E 3 $$
;
/.L=l i=l
,._., n,
j=l,Z
where fj~ is @VHI be 4P p =1,2 ,...,
i=l,2,...,qk;
Since
+$f
phase factor
N is a rea1, ]~!~~.‘_‘::4,
Iinearly
N.
(5)
independent set we find that ~j is real apart from a trivial
exp (iaj) if and only if tg 0.j = Im $/Re
_$
i= 1,2,...,+
;
;
/.J =1,2 ,...,
N.
(6)
(Re and Im mean the reai and imaginary parts, respectively;) The construction of a real total wave function from an antisymmetrized product of orbits& $‘j (i = I, 2, - . . , n), however, does not require (6) as a necessary condition. In ref. [Z] it is proved that the tota.I determinant function is real (apart from a p&+se factor) if and only if there exists a unitary transformation among its natural spin-orbitals that makes them real. In the case of pure spin-orbitals, we thus consider transformations between the *j’s of d spin and the J/i’s of fi spin separately. The spin functions Q and B are here considered as real. Denoting the number of orbitals with a! and B spin, by n, and no, respectively, we find that the determinant is real if and only if in the set {Re@ja,ImlC/jc,) there are n,, and in the set {Reqj6, Im +j6>, n 6 linearly independent functions. The possibility of a transformation to real orbitals means that no restriction is composed by condition (6). In the case of complex hybrids, on the other hand, (6) cannot generally be fulfilled. Rewriting (3) as N qf~ Gj = [(Regjt + i Imgjg) Rehg + (i Regjs -Imgjs) ImI$] 9 0) ,u=l k=l
c c
and assuming that Re h$ and Im$, obtained % "j = Imgjs/Regjx
p = 1,2, . . . , N, k = 1,2,. - . , qp, are all linearly
= - RegJ$/Imgj$
or p
=
;
1,2,...,N;
p = 1,2,. . .,N; k=l,2,...,qp;
independent,
k= I,2 ,..., j=l,2
(8)
qu , ,...,
it is
n,
(9)
which indeed is a contradiction. The additional feature of o’ur analysis is that we get a complex determinant wave function. The underlying assumption is of course that the real and imaginary parts of the hybrids should be linearly independent. In most practical cases this is not true, but t~ldeterm$ant is still complex if the number of Iinearly independent functions in the set {Rehg ,Imh$)k:lf : : : iqP exceeds the smaller of n@ and ns. It is well known that the total wave function for a system described by a real Hamiltonian, should be real (or at least without restriction could be chosen so). Otherwise, the total momentum is not conserved and the system is not invariant against reversal of time [3]. Therefore the method of complex hybridization, which removes the anguIar restrictions connected with conventional hybridization, 316
Volume 3, number5
CHEMICALPHYSICSLETTERS
May 1969
instead seems to bre& the time reversal symmetry of the system. This difficulty is in fact nothing but a consequence of the old symmetry dilemma [4]: i.e. the incompatibility between the Rartree-Fock scheme and the symmetry requirements. In the Hartree-Fock scheme or various extensions of it [5] the variation principle is applied to the antisymmetrized product of the Qj multiplied with the appropriate spin functions. From the Variational point of view (6) is a constraint, which may raise the energy expectation value [2]. The most general treatment in this context will be to vary the real and the imaginary parts separately, which is in agreement with the idea bebind the Projected Hartree-Fock method [g]. The above statement is due to a theorem which states that either the real or the imaginary part of a complex wave function optimizes its expectation value [Z]. The determinant based on complex Hybrids thus consists of two real components, one corresponding to a lower and the other to a higher energy expectation value (except in the degenerate case) compared to the average value obtained from the complex determinant. The correct wave function of the system is consequescly obtained from a component antiysis of the original determinant constructed from the hybrids h#.
3. THE YB METHOD In the valence bond scheme we construct the singlet functicn for a bond between centers g.and v from the appropriate hybrids hp(l) and l?(Z) by perfect pairing of electrons 1 and 2. As a result we get a valence bond geMi.tld of the following form G91,2)
= {hCc(1)hu(2)+h”(l)h~(2)}
The total valence bond singlet for the appropriate
,pvB = -&
(a1S2;1a2)
.
(10)
structure is then given by the expression
. .Gpmum(n-l,n))
(-1)p~{Gp1u1(1,2)Gp2u2(3,4).
(tr=2m)
.
(11:
‘P
Note that each GpV(l, 2) May be obtained as a projection of the zero-spin component of the antisymmetrized product of the orbit&s h~(l)crlh”(2)/32. The wave function for the whole system till then be obtained by a superposition of the correct number of valence bond singlets given, for inStanCe, by Rumer’s rule [6]. For the valence bond singlet it is easy to see that each geminal G”(l, 2) in general mLSt be Complex. In order to realize this we make the transformation (1~I-r +IzV)/d2(1 +ReSp”)
= c1;
(h’l -hv)/J2(1
-ReSn”)
= b ;
sp”
= (iP pzV) ;
(12)
and the subsequent expression for G p’(i, 2) then reads in normalized form (note that (a lb) + 8 ia) = 0, and that it is possible to choose the relative phases of hp and h” SO that (a 16) = 0, i.e. scru = @ ‘?‘)-
aiP~~‘1a2 =--&{c~
Gpu(1,2)=~{h'l(l)~t"(2)+h"(l)h'(2)} c
=N(l+ReS”) a
*,
det(aa,a/3)
- cb det(hba,b&)i ;
cb = N(l-ReSCCu)
The correspondence occupation numbers (eigenvalues to the one-electron GKu(l, 2)) thus will be, provided ScsV from now on is chosen real,
.
density Matrix p based
2 2 “b = =b 9 % = ca ; where both na and nb are doubly degenerate. We thus have for the density Matrix ~(1,
1’) =
2
sGp”(l,
2){Gpu(l’,
2)}* dx2 = (alai
+BlPi){n,a(
rl)(a(
~~;)*+nbb(rl)(b(ri))“)
OIL
(14)
;
95)
Tr(p)=2, and assuming the real and imaginary parts of fip and h” to be linearly independent, which also appJ.fes to a, a*, b and b*, we note that Gp”(l, 2) in general must be complex. If not; we get 317
Volume 3. nzuher 5
CREMICAL PFXY8ICSLETTERS p - p* = (aa? +&3’) {na det (aa*) + “6 det (bb*))
May 1969
;
(16)
which implies that a, a*, b and b* cannot be linearly independent. Note that the case a* = b has been discussed in ref. [?I. Since thz total system is considered to be invariant under time reversal, it is therefore necessary to decompose the total wave function in its complex components. A simpler procedure, but more restrictedd, 1s obtained from the separation of each geminal into its real aud imaginary parts before the antisymmetrization is performed. This, however, implies tkt we require all the bonds to be symmetric with respect to reversal of time, which by no means should be considered as a necessary condition. The total wave function based on the geminals (10) will therefore not iu general be real, which means that the final wave function for the total system should be obtained by a real projection of the valence bond singlet. ACKNOWLEDGEMENTS We are very grateful to professor P-0. Ltlwdin for his support and inspiration. to Professor Y.&r-n and Dr. R. Manne for a critical reading of the manuscript.
We are also indebted
REFERENCE8 [l] O.Mtitensson and V.C)hrn, Theoret. chim. Acta (Berlin) 9 (1967) 133; 0. Mtlrtensson, Uppsala Quartum Chemistry Group. Prelimtiary Research Report No. 232. January 1969. 121E.BrS.ndas. J. Mol. SPectry. 37 (1968) 236. [3] H.A.Kramers. Koninkl. Ned. Akzd. u’etenschap. Proc. ser. B 33 (1930) 959. [4] J.C.Slater. Phys. Rev. 35 (1930) 509. [5] P.O.Liiudin, in: Quantum theory of atoms, molecules and the solid state, Slater volume. ed. P.O.Ltiudin (Academic Press, New York, 1966) p. 601. [6] G,Rumer, Gijttingen Nachrichten. (1932) 372. [7j P.O. Liiwdin and H. Shull. Phys. Rev. 101 (1956)
318
1730.
3. number
5
WAVE
CWMICAL
FUNCTION
PHYSICS
CONSTRUCTED E. BR&DAS
Quantum Chemistry
Group. Received
LETTERS
FROM
May 1969
COMPLEX
HYBRIDS
and 0. MARTENSSON Uppsakz University. 19 February
Uppsaka. Sweden
1969
The influence on the total wave function of an atomic basis consisting of complex hybrids is cousidered. The complex character of the wave function in the MO-LCAO scheme ss well as in the VI3 method is derived, and its consequences with respect to time reversal symmetry are discussed.
1. INTRODUCTION Hybrids formed by linear combination of atomic orbitals of s- and p-type are very useful at least for a qualitative understanding and explanation of the chemical bond and other concepts related to this, such as bond direction and bond angle. The bond is described by so-called perfect pairing of two electrons, one from each of two hybrids creating the bond. Conventional hybridization, however, has the limitation that the valence angle definition, which is based on the orthogonality between the hybrids, does not permit vr-lence angles less than 90°. Moreover, equivalency as regards contents of s-orbital exists only for a true tetreonal configuration. A redefinition of the bond (valence) directi= concept and the permission of complex elements in the hybridization matrix (which we consider unitary unless otherwise stated) lead to a procedure, complex hybridization [l], which removes several of the difficulties connected with conventional hybridization, but introduces other problems and aspects which are to be discussed. Using +&e notation of ref. [l], the hybrid of center ,u, numbered k, will be expressed by h;
where 4;
=
ztlah C#J;;
g = 1,2 ,...,
is an atomic orbital of center p, for instance described
N, by
(2)
andAC’={aklisth e unita.y hybridization matrix of the same center, q finally depending on the special choice of ybrklizafion (for orbitals of s- and p-type 4 = 4). The elements a - may be complex and are obtained such that they fulfill given-bond-angle conditions for the atom (ten$ er) & From this it follows that, when complex coefficients occur in the hybrid, these can not in general he factored out, merely forming a phase factor of an otherwise real hybrid. One can, however, without restriction in the first column of AN will be real and positive. choose the phase so, that the element {a[l} A total trial wave function of a molecule may be constructed from the hybrids (1) =h different ways. We can form molecular orbitals by linear combinations of the hybrids, and thereafter take the determinant of the product of MO’s. More natural in this context is to form valence bond gemin&s from the appropriate hybrids and then antisymmetri:ce the whole geminal product. In the following we wiU commat on some aspects on the wave function built up from complexhybrids.
315
Volume
2. THE
3, number
MO-LCAO
5
CHEMICAL
PHYSICS
LETTERS
May 2969
METHOD
In the molecular-orbital method we consider the complex hybrids {I#) in (1) as the basis functions for the total wave function of the system considered. The molecular orbitals are then given by N
qlJ
N
C qp =fl;
j=l,2
,...,
(3)
n,
/l=l
where N is the number of centers, n is the number of valence electrons, and qp indicates the choice of hybridization on atom p_ From (1) we obtain ~j expressed in the atomic orbitak $B. *j
=
E 3 $$
;
/.L=l i=l
,._., n,
j=l,Z
where fj~ is @VHI be 4P p =1,2 ,...,
i=l,2,...,qk;
Since
+$f
phase factor
N is a rea1, ]~!~~.‘_‘::4,
Iinearly
N.
(5)
independent set we find that ~j is real apart from a trivial
exp (iaj) if and only if tg 0.j = Im $/Re
_$
i= 1,2,...,+
;
;
/.J =1,2 ,...,
N.
(6)
(Re and Im mean the reai and imaginary parts, respectively;) The construction of a real total wave function from an antisymmetrized product of orbits& $‘j (i = I, 2, - . . , n), however, does not require (6) as a necessary condition. In ref. [Z] it is proved that the tota.I determinant function is real (apart from a p&+se factor) if and only if there exists a unitary transformation among its natural spin-orbitals that makes them real. In the case of pure spin-orbitals, we thus consider transformations between the *j’s of d spin and the J/i’s of fi spin separately. The spin functions Q and B are here considered as real. Denoting the number of orbitals with a! and B spin, by n, and no, respectively, we find that the determinant is real if and only if in the set {Re@ja,ImlC/jc,) there are n,, and in the set {Reqj6, Im +j6>, n 6 linearly independent functions. The possibility of a transformation to real orbitals means that no restriction is composed by condition (6). In the case of complex hybrids, on the other hand, (6) cannot generally be fulfilled. Rewriting (3) as N qf~ Gj = [(Regjt + i Imgjg) Rehg + (i Regjs -Imgjs) ImI$] 9 0) ,u=l k=l
c c
and assuming that Re h$ and Im$, obtained % "j = Imgjs/Regjx
p = 1,2, . . . , N, k = 1,2,. - . , qp, are all linearly
= - RegJ$/Imgj$
or p
=
;
1,2,...,N;
p = 1,2,. . .,N; k=l,2,...,qp;
independent,
k= I,2 ,..., j=l,2
(8)
qu , ,...,
it is
n,
(9)
which indeed is a contradiction. The additional feature of o’ur analysis is that we get a complex determinant wave function. The underlying assumption is of course that the real and imaginary parts of the hybrids should be linearly independent. In most practical cases this is not true, but t~ldeterm$ant is still complex if the number of Iinearly independent functions in the set {Rehg ,Imh$)k:lf : : : iqP exceeds the smaller of n@ and ns. It is well known that the total wave function for a system described by a real Hamiltonian, should be real (or at least without restriction could be chosen so). Otherwise, the total momentum is not conserved and the system is not invariant against reversal of time [3]. Therefore the method of complex hybridization, which removes the anguIar restrictions connected with conventional hybridization, 316
Volume 3, number5
CHEMICALPHYSICSLETTERS
May 1969
instead seems to bre& the time reversal symmetry of the system. This difficulty is in fact nothing but a consequence of the old symmetry dilemma [4]: i.e. the incompatibility between the Rartree-Fock scheme and the symmetry requirements. In the Hartree-Fock scheme or various extensions of it [5] the variation principle is applied to the antisymmetrized product of the Qj multiplied with the appropriate spin functions. From the Variational point of view (6) is a constraint, which may raise the energy expectation value [2]. The most general treatment in this context will be to vary the real and the imaginary parts separately, which is in agreement with the idea bebind the Projected Hartree-Fock method [g]. The above statement is due to a theorem which states that either the real or the imaginary part of a complex wave function optimizes its expectation value [Z]. The determinant based on complex Hybrids thus consists of two real components, one corresponding to a lower and the other to a higher energy expectation value (except in the degenerate case) compared to the average value obtained from the complex determinant. The correct wave function of the system is consequescly obtained from a component antiysis of the original determinant constructed from the hybrids h#.
3. THE YB METHOD In the valence bond scheme we construct the singlet functicn for a bond between centers g.and v from the appropriate hybrids hp(l) and l?(Z) by perfect pairing of electrons 1 and 2. As a result we get a valence bond geMi.tld of the following form G91,2)
= {hCc(1)hu(2)+h”(l)h~(2)}
The total valence bond singlet for the appropriate
,pvB = -&
(a1S2;1a2)
.
(10)
structure is then given by the expression
. .Gpmum(n-l,n))
(-1)p~{Gp1u1(1,2)Gp2u2(3,4).
(tr=2m)
.
(11:
‘P
Note that each GpV(l, 2) May be obtained as a projection of the zero-spin component of the antisymmetrized product of the orbit&s h~(l)crlh”(2)/32. The wave function for the whole system till then be obtained by a superposition of the correct number of valence bond singlets given, for inStanCe, by Rumer’s rule [6]. For the valence bond singlet it is easy to see that each geminal G”(l, 2) in general mLSt be Complex. In order to realize this we make the transformation (1~I-r +IzV)/d2(1 +ReSp”)
= c1;
(h’l -hv)/J2(1
-ReSn”)
= b ;
sp”
= (iP pzV) ;
(12)
and the subsequent expression for G p’(i, 2) then reads in normalized form (note that (a lb) + 8 ia) = 0, and that it is possible to choose the relative phases of hp and h” SO that (a 16) = 0, i.e. scru = @ ‘?‘)-
aiP~~‘1a2 =--&{c~
Gpu(1,2)=~{h'l(l)~t"(2)+h"(l)h'(2)} c
=N(l+ReS”) a
*,
det(aa,a/3)
- cb det(hba,b&)i ;
cb = N(l-ReSCCu)
The correspondence occupation numbers (eigenvalues to the one-electron GKu(l, 2)) thus will be, provided ScsV from now on is chosen real,
.
density Matrix p based
2 2 “b = =b 9 % = ca ; where both na and nb are doubly degenerate. We thus have for the density Matrix ~(1,
1’) =
2
sGp”(l,
2){Gpu(l’,
2)}* dx2 = (alai
+BlPi){n,a(
rl)(a(
~~;)*+nbb(rl)(b(ri))“)
OIL
(14)
;
95)
Tr(p)=2, and assuming the real and imaginary parts of fip and h” to be linearly independent, which also appJ.fes to a, a*, b and b*, we note that Gp”(l, 2) in general must be complex. If not; we get 317
Volume 3. nzuher 5
CREMICAL PFXY8ICSLETTERS p - p* = (aa? +&3’) {na det (aa*) + “6 det (bb*))
May 1969
;
(16)
which implies that a, a*, b and b* cannot be linearly independent. Note that the case a* = b has been discussed in ref. [?I. Since thz total system is considered to be invariant under time reversal, it is therefore necessary to decompose the total wave function in its complex components. A simpler procedure, but more restrictedd, 1s obtained from the separation of each geminal into its real aud imaginary parts before the antisymmetrization is performed. This, however, implies tkt we require all the bonds to be symmetric with respect to reversal of time, which by no means should be considered as a necessary condition. The total wave function based on the geminals (10) will therefore not iu general be real, which means that the final wave function for the total system should be obtained by a real projection of the valence bond singlet. ACKNOWLEDGEMENTS We are very grateful to professor P-0. Ltlwdin for his support and inspiration. to Professor Y.&r-n and Dr. R. Manne for a critical reading of the manuscript.
We are also indebted
REFERENCE8 [l] O.Mtitensson and V.C)hrn, Theoret. chim. Acta (Berlin) 9 (1967) 133; 0. Mtlrtensson, Uppsala Quartum Chemistry Group. Prelimtiary Research Report No. 232. January 1969. 121E.BrS.ndas. J. Mol. SPectry. 37 (1968) 236. [3] H.A.Kramers. Koninkl. Ned. Akzd. u’etenschap. Proc. ser. B 33 (1930) 959. [4] J.C.Slater. Phys. Rev. 35 (1930) 509. [5] P.O.Liiudin, in: Quantum theory of atoms, molecules and the solid state, Slater volume. ed. P.O.Ltiudin (Academic Press, New York, 1966) p. 601. [6] G,Rumer, Gijttingen Nachrichten. (1932) 372. [7j P.O. Liiwdin and H. Shull. Phys. Rev. 101 (1956)
318
1730.