How Accurate is the Spin-Coupled Valence Bond Wavefunction for H3?
In order to discuss this topic we must explain how in principle one can approximate
the exact three electron wavefunction to arbitrary accuracy using functions which can be
stored in a computer. The spinless one electron Hilbert space is L^2(R^3), an infinite
dimensional vector space. The first step is to introduce a finite basis set B,
consisting of a linearly independent subset of L^2(R^3). By enlarging B appropriately one can
approximate any function in L^2(R^3) by a function in V=span(B) with arbitrary accuracy.
If the functions in B are not orthogonal with respect to the inner product from L^2(R^3)
then some orthogonalization process (e.g. Gram-Schmidt) can be performed on the members of B
to yield an ordered orthonormal basis set B' so that span(B')=V. Let B'={f_1,f_2,...,f_N}.
The approximate one electron (with spin) Hilbert space is W = V tensor C^2.
The approximate three electron Hilbert space is W /\ W /\ W, the antisymmetric tensor
product of W with itself three times.
Theorem
The most general three electron wavefunction in W /\ W /\ W
(with total squared spin of 3/4 and total z-component of spin 1/2) is of the form:
sum_{1 <= i < j < k <= N}
{
a_{i,j,k} A[ f_i(r1)*f_j(r2)*f_k(r3)*Kotani1(s1,s2,s3) ] +
b_{i,j,k} A[ f_i(r1)*f_j(r2)*f_k(r3)*Kotani2(s1,s2,s3) ]
}
+sum_{1 <= i , k <= N , i not= k}
{
c_{i,k} A[ f_i(r1)*f_i(r2)*f_k(r3)*Kotani2(s1,s2,s3)]
},
where a_{i,j,k}, b_{i,j,k}, and c_{i,k} are real constants, and
Kotani1(s1,s2,s3)=[2a(s1)a(s2)b(s3)
-a(s1)b(s2)a(s3)
-b(s1)a(s2)a(s3)]*6^(-1/2),
Kotani2(s1,s2,s3)=[a(s1)b(s2)a(s3)
-b(s1)a(s2)a(s3)]*2^(-1/2);
for more info.
All these terms are orthogonal to each other, and no term is identically zero if its coefficient
is nonzero.
A minimal energy wavefunction of this form is called the Full CI (FCI) wavefunction
associated to the basis set B'.
Each term (not including the coefficient) is called a Configuration State Function (CSF).
Each CSF is a linear combination of Slater determinants, which are expressions
such as A[ f_i(r1)a(s1) f_j(r2)a(s2) f_k(r3)b(s3) ];
such determinants need not be spin eigenfunctions.
This CSF basis of a spin eigensubspace of W /\ W /\ W gives us some idea of the complexity of
exact wavefunctions and helps us to understand why they are so difficult to visualize or
intuitively understand.
By contrast the spin-coupled valence bond (scvb) wavefunction is of the form
phi(r1,r2,r3,s1,s2,s3)=A[ psi1(r1)*psi2(r2)*psi3(r3)*{c_1*Kotani1(s1,s2,s3)+c_2*Kotani2(s1,s2,s3)} ],
where the orbitals psi1, psi2, and psi3 are in V, but not required to be orthogonal.
psi1, psi2, psi3, c1, and c2 are chosen so as to minimize the energy.
It is possible to choose B' so that span{psi1,psi2,psi3}=span{f_1,f_2,f_3}. Thus
psi1=d11*f_1+d12*f_2+d13*f_3,
psi2=d21*f_1+d22*f_2+d23*f_3,
psi3=d31*f_1+d32*f_2+d33*f_3,
where d11,...,d33 are constants. Therefore phi is equal to
sum_{1 <= i , j , k <= 3}
{
d1i*d2j*d3k A[ f_i(r1)*f_j(r2)*f_k(r3)*{c1*Kotani1(s1,s2,s3)+c2*Kotani2(s1,s2,s3)} ]
}.
This sum has 54 terms (27 choices of (i,j,k), two choices of spin function for each)
but not all of these are nonzero or unique. There are 6 terms where i=j=k, all of which vanish.
There are 12 terms where i,j,k are all distinct. Those with i > j or j > k can (at the cost of
changing the values of c1, c2) be rewritten in terms of the two terms with i < j < k.
The 36 remaining terms break down as follows: 12 where i=j, 12 where i=k, and 12 where j=k.
The ones with i=k or j=k can be rewritten in terms of the ones with i=j. Of these 12 (where i=j)
half (the ones involving Kotani1) vanish, leaving 6 distinct nonvanishing ones.
Thus phi equals
a_{123} A[ f_1(r1)*f_2(r2)*f_3(r3)*Kotani1(s1,s2,s3) ]...(//\)... +
b_{123} A[ f_1(r1)*f_2(r2)*f_3(r3)*Kotani2(s1,s2,s3) ]...(/\/)... +
c_{12} A[ f_1(r1)*f_1(r2)*f_2(r3)*Kotani2(s1,s2,s3) ]...(2/0)... +
c_{13} A[ f_1(r1)*f_1(r2)*f_3(r3)*Kotani2(s1,s2,s3) ]...(20/)... +
c_{21} A[ f_2(r1)*f_2(r2)*f_1(r3)*Kotani2(s1,s2,s3) ]...(/20)... +
c_{23} A[ f_2(r1)*f_2(r2)*f_3(r3)*Kotani2(s1,s2,s3) ]...(02/)... +
c_{31} A[ f_3(r1)*f_3(r2)*f_1(r3)*Kotani2(s1,s2,s3) ]...(/02)... +
c_{32} A[ f_3(r1)*f_3(r2)*f_2(r3)*Kotani2(s1,s2,s3) ]...(0/2)...,
for an appropriate choice of the eight constants a_{123}, b_{123}, c_{12},...,c_{32},
all of which can be written as various expressions involving d11, ..., d33, and c1, c2.
We have followed each CSF with "...label..." where label denotes its Molpro
occupation symbol.
An approximate wavefunction with appropriately minimized values of the eight constants, and
optimal choices for f_1, f_2, f_3 is called a CAS(3,3) wavefunction.
Notice that the Cas(3,3) wavefunction has the same CSFs as a FCI wavefunction with N=3.
These same CSFs remain present in the FCI wavefunction for larger values of N,
but of course their coefficients might be changed.
The usual algorithm for computing the Cas(3,3) wavefunction starts with the basis set B
and yields an orthonormal basis set B' and a set of the eight coefficients so as to minimize
the energy. The energy does not depend on the choice of f_4, ..., f_N but only on f_1, f_2, f_3,
and the eight coefficients. Nevertheless f_4, ..., f_N are available after the algorithm is done.
Empirical Fact: For essentially all practical purposes the Spin-Coupled Valence Bond
wavefunction is equal to a Cas(3,3) wavefunction.
Thus we are interested in the question of how well the FCI wavefunction is approximated
by a Cas(3,3) wavefunction. We can break the FCI wavefunction into two mutually orthogonal parts:
FCI wavefunction = Cas(3,3) terms (i.e. those with both i and k <= 3) + other terms.
So 1=|| FCI wavefunction ||^2 = || Cas(3,3) terms ||^2 + || other terms ||^2.
Define C_0 = || Cas(3,3) terms || = [ 1 - || other terms ||^2 ]^(1/2) < 1.
The Cas(3,3) terms of the FCI wavefunction will not be equal to the Cas(3,3) wavefunction.
Since || Cas(3,3) wavefunction || = 1, at the very least we expect to have to
multiply the constants a_{1,2,3}, b_{1,2,3}, c_{1,2},...,c_{3,2} in the Cas(3,3) wavefunction
by a constant C_0<1 to get the constants in the Cas(3,3) terms of the FCI wavefunction with
the same names. This simple relationship will not in general be exactly true,
but it is often a pretty good approximation. The FCI coefficients of triple excitations,
i.e. CSFs
with both i and k >= 4, are usually very small.
The multireference CI (mrci) wavefunction takes the following form:
mrci wavefunction
= c_0* Cas(3,3) wavefunction
+ non Cas(3,3) CSFs from the FCI wavefunction with at least one of i, k <= 3.
In this expression c_0 is not the same as C_0 we defined earlier, although it plays a similar role.
It is customary to use the orthonormal basis set B'={f_1,f_2,f_3,f_4,...,f_N} which is the result
of the algorithm which computes the Cas(3,3) wavfunction when finding the mrci wavefunction.
The constants c_0, a_{i,j,k}, b_{i,j,k}, c_{i,k} (excluding those with i and k <=3 or i and k >=4)
are optimized so as to minimize the energy. The orthonormal basis set B' is not changed during
the minimization process. Since the minimization for the mrci wavefunction has fewer free
variables than the minimization for the FCI wavefunction, these two wavefunctions will not be
exactly equal. However they are pretty close. The mrci wavefunction is normalized, so that
c_0 = [ 1 - || non Cas(3,3) non triple excitation CSFs from the FCI wavefunction ||^2 ]^(1/2).
Defining Theta
A direct measure of the amount of change between the Cas(3,3) wavefunction and the mrci wavefunction
is the angle theta between these two real-valued normalized elements of W /\ W /\ W.
cos( theta ) = [Cas(3,3) wavefunction] dot [mrci wavefunction]
= [Cas(3,3) wavefunction] dot [ c_0* Cas(3,3) wavefunction + other CSFs from the mrci wavefunction ]
= c_0, so theta = cos^{-1} c_0.
For example if c_0 = 0.992 (a typical value) then theta = 7.25 degrees, a fairly small angle.
We plan to report the angle theta whenever we compute a scvb wavefunction as a rough
way of assigning a confidence level to the intuitive scvb picture of the FCI wavefunction.
This assumes the validity of the empirical fact, the near equality of c0 times the Cas(3,3) coefficients
to the FCI coefficients of those CSFs, and the negligibility of triple excitations.
It is also natural to report the amount that the energy can be reduced by changing from the
visual simplicity of the scvb wavefunction to the essentially incomprehensible
mrci wavefunction. These two measures of accuracy are complementary, and need not be correlated,
since the Hessian of the energy function at its minimum point on the unit sphere in W /\ W /\ W
is not known or fixed.
Question: Is there a way to expand the FCI wavefunction, where the scvb
wavefunction is the first term, and where each of the `higher order terms' have some sort
of visual interpretation? For example, one might posit nonorthogonal orbitals psi4, psi5, psi6
and additional small constants C1', C2', C3', and new spin functions spin1, spin2, spin3 such that:
FCI wavefunction =
c0' A[ psi1 psi2 psi3 c1*Kotani1+c2*Kotani2 ] (...scvb term...) +
C1' A[ psi4 psi2 psi3 spin1 ] (...excitation of psi1 to psi4, spins recoupled...) +
C2' A[ psi1 psi5 psi3 spin2 ] (...excitation of psi2 to psi5, spins recoupled...) +
C3' A[ psi1 psi2 psi6 spin3 ] (...excitation of psi3 to psi6, spins recoupled...) + `higher order terms'.