Constraints and thev2n+1 theorem

Volume 70, number 2

CONSTRAINTS

CHEMICAL

AND THEv2”+l

PHYSICS

LETTERS

I March I980

THEOREM

Saul T. EPSTEIN Physics Department, University of Wtsconsin. Maduon, Wasconan 53706. US4 Received

12 December

A general formula

1979

IS derived from which, knowing the variatIona

parameters

and Lagrange multipliers through order Ym

one can calculate the varlational energy through order .*n+l_

The y2n+1 theorem [ 1, sectron 261 provides a general prescriptron for calculating the variational energy through order v2n+1 given the variatronal parameters through order Y”, and is a very fundamental and useful theorem. However the theorem m ref. [I] does not allow for the possibrlity that the varrational equations may contam Lagrange multipliers because of constraints on the tnal functrons *. In this note we will denve a more general theorem which holds also in the presence of constramts, and which provides a prescription for calculating the energy through order v2n+l given the variational parameters and the Lagrange multiphers through order Y”. We NIU denote the variational parameters by A, and for simphcity of presentation we v&l assume them to be real numerical parameters. (Tlus is no restriction in principle. More importantly, in our final formula the nature of the A, wdl be irrelevant.) If we use a _ to distinguish optrmal values, then the optimal energy wrll have the form l? = E(i

, v).

Suppose

now that

,&=A;-+O(v”+l)

(0

and define I? kE(i,v),

(3)

then we have that ,k = I? + c However

(Al - ,di) &!?/i3ai f (S(V~~~~).

the variational

equations

are

where the Z, are Lagrange multipliers, and where the es are the constramts, i.e. the constraint equations are es = C,(A^, v) = 0. Therefore

(6)

we can write (4) as

fZ = k - q

C

S

{(Aoi - A-i) aks’,/aAmi f 6(G+2),

where we have also mtroduced such that

(2)

* Nevertheless the theorem of ref. [ l] was successfully apphed SCF theory 111section 28 of ref. [ 11. However m this case it IS easy to see that the derivation in ref. [l] IS vahd even with the constraints Namely m this theory onedoeshave@&(ff-E)$)+($,(H-E)S$)=Ofor all 6 I$, and not just for those which satisfy the constraints.

(7)

is which should be

1”,= r;, +,(v,+t).

(8)

We now defme es by cs = es&

to unrestncted

I

v)_

(9)

Then using (6) it follows that Z’S = C (pi - Ai) a@& i

whence

+ O(V~~*~),

we can write (7) as 3LE

Volume 70. number 2

CHEMICAL PHYSICS LEITERS

(11-I which

is our fiial result. From it one can evidently cdculate the energy coqectly through order ~2’~+1 by knowmg the variational parameters,znd the Lagrange muItipliers through order S.

312

1 March 1980

Reference [I 1 S.T. Epstem, The variation method in quantum chemlstw (Academic Press, New York, 19741, and references therem.