(Ho -o)Oo(l) - Europe PMC

10 downloads 0 Views 448KB Size Report
(Ho -o)Oo(l) + (V -o(1))60 = 0. (2). (Ho - g0)L0(2) + (V - g0(l))J'0(l) = &(2)o. (3) .... single-valued at a point (Xo, Yo, Z0) and at least one of them is nonzero at this.
AN ITERATIVE PROCEDURE FOR THE SOLUTION OF PERTURBATION EQUATIONS* BY DONALD K. HARRISSt AND JOSEPH 0. HIRSCHFELDER UNIVERSITY OF WISCONSIN THEORETICAL CHEMISTRY INSTITUTE, MADISON

Communicated December 15, 1967

An often-used procedure in the solution of differential equations is an iterative one in which the first approximation retains only those terms expected to be large in the region of interest. This type of procedure has been used in the solution of differential equations arising in quantum theory such as in the work of Ovchinnikov and Sukhanovl on H2+ in which case they were able to separate variables and treat the separate differential equations. If the usual Rayleigh-Schr6dinger perturbation equations

(Ho- o) O = 0 (Ho -o)Oo(l) + (V -o(1))60 = 0 (Ho - g0)L0(2) + (V

-

g0(l))J'0(l)

= &(2)o

(1) (2) (3)

are assumed to have solutions of the Dalgarno-Lewis form2 for o(') #o2, ...,

lo(l)

=

4o(2) =

Fjo G('o

(4) (5)

and are written in the form

(6) V. (V/02VF) = 20o(V -o(1))0 g(l)),0(l) -28 (2),0(I), _2(7) V. (4o2VG) = 24'(Van iterative scheme for their solution is suggested based on the following arguments, illustrated with the first-order equation, equation (6). If we expand the left-hand side of equation (6)

'02V * VF + V002 * VF, and consider regions where #o2 is very small, the second term might be expected to dominate. Although no a priori statement can be made about regions where the arguments are small, certainly for large values of the arguments where #0o2 might be expected to behave as r' exp (-ar) and F as rm the second term will dominate. This suggests an iterative procedure in which the first approximation neglects 00o2V * VF, (8) V#02*VFo = 2#o(V - Molo = fJ and higher approximations to F are determined from 319

PHYSICS: HARRISS AND HIRSCHFELDER

320

PROC. N. A. S.

(9) 02V * VFj. Boundary and normalization conditions on the successive F. or G, would be those normally associated with the various orders of perturbed wave functions and the Dalgarno-Lewis form3 (1Oa) Fj4o, #o2VFj = 0 on the boundaries * VFj+1 = f -y

(#o I Fj4o) = 0,

(lOb)

or

Gj,'o, 'o2VGj = 0 on the boundaries 2(ol Gj4,o) + (ko(l)I 6o(1)) = 0.

(hla) (1 lb) Criteria for convergence of Fj to F can be based on Fi+i = Fj or if the sequence does not appear to be terminating after a finite number of steps, one might consider the numerical convergence of a sequence based on the Hylleraas variational principle for the second-order energy4

go(2)

(P(')HHo - 8o l;()) + (4(1)I V' ,#o) + (4c01 VI I4;(1))

) So(2) where ;(') is arbitrary and VI = V -Go(1). That is, if we define

&o, (2)

=

(F40oI Ho -So| Fj40) + (F40o I V'j|

o) +

(Vo|I VII Fj0),

(12)

(13)

the numerical convergence of the &oi(2) may yield a satisfactory convergence criterion for an approximate F. Similar upper bound variational principles for 8O(4) can be used as a convergence criterion for an approximate G. A particular virtue of this iterative scheme is that the equations necessary to solve are linear first-order inhomogeneous partial differential equations which can be treated by the method of characteristics. A short review of this method may be useful at this point. Method of Characteristics.5-The method of characteristics is most easily discussed in the case of two independent variables. If we let x and y be the two independent variables and z the dependent variable, then, if p and q are the partial derivatives of z with respect to x and y, respectively, a general equation of this type is given by tp + ?7q =(14) where I, 7, and r are functions of x, y, and z. If z = F(x,y) is a solution of this equation, we must have (6F) + (F) = (15) for all values of x and y. For the case of two independent variables the function F would determine a surface, called the integral-surface of the partial differential equation. If we now determine the solutions of the system of ordinary differential equations

VOL. 59, 1968

PHYSICS: HARRISS AND HIRSCHFELDER

321

dy

(16)

dx

dz

and solve these solutions for the constants of integration U(xyz) = a, V(xyz) = S, the functions U and V represent a two-parameter family of curves in space which are known as the characteristics of the system. If t, aq, and t exist and are single-valued at a point (Xo, Yo, Z0) and at least one of them is nonzero at this point, then one and only one characteristic passes through that point. The importance of the characteristics lies in the fact that it can be shown that: (a) if an integral-surface passes through (X0, Yo, Z0), it contains the characteristic through that point, and, more important, the converse, that (b) in general every surface arising as a loctis of characteristic curves is an integral-surface of the partial differential equation. Thus if a relationship is set up between a and f, say A(a,f3) = 0, the corresponding relation, A(U,V) = 0, between U and V is an equation to the integral-surface and this equation for arbitrary A is the general solution to the partial differential equation. If we first define If o2) (17)

where hi is the metric scale factor for the orthogonal curvilinear coordinate qj, and symbolize the inhomogeneous portion as g = g(qiq2,... ,qn), the equations involved in this work can be written as

Ofl

=)=9

(18)

The associated set of simultaneous ordinary differential equations then would be dql dq2 dqn dF (19) 4i

Oin

Ot2

9

where neither the As nor g involves F. If the solutions of this set of ordinary differential equations, solved for the constants of integration, are assumed to be Sl(qlq2, ..., qn,,F) = al,

S2(qlq2,

...,q,1,F)

=

a2,

Sn-1(ql,)q2 ..., qF)

=

an-ly

we see that the general solution to the partial differential equation is any arbitrary function (20) A(SjjS2 * . Sn-1) = 0. To determine the desired form of A we have boundary conditions on F, which

322

PHYSICS: HARRISS AND HIRSCHFELDER

PROC. N. A. S.

are sufficient in most cases, and can establish additional criteria such as that A yields the best upper bound by the Hylleraas variational principle. Examples.-The nature of the iterative scheme can be illustrated with a few simple examples: (A) Two-dimensional harmonic oscillator coupled by V = kxy/3: For this system with Uo = k(x2 + y2)/2 we have

i02

1)2 eP(V2 + i'!

= (

(21a)

So(1) = 0 f = 2(V -g0(l))P02 = 2/3 kxy4/o2

V4102 VF

=

-2&xo2 (a

-

20Yo2 (a)

(21b) (21c)

(21d)

Indicial equation: (22a)

V o2 VFo = f

or

(aFo) +

( Fo) =

-kxy/3f3,

(22b)

which has the solution, obeying boundary conditions, (22c) Fo= -kxy/60. The next approximation, F1, is found to be identical with Fo and therefore

OM1) = Fjto

=

(23)

-(kxy/6j3)#o.

This function yields the correct second-order energy 8O(2)

=

VI i/om) (,0l'

=

-

1

fk

and, of course, the third-order energy is zero for symmetry reasons. (B) Ground-state hydrogen atom in a uniform electric field: The perturbation may be taken as -Z for unit field strength and we know 2 =6

1

e-2r

o(1)= 0.

(B.1) Determination of Oo(l) =F4o: f = 2(V - go(l))t1o2 = -2r cosOJt2 Vo2 VF = -2#o22 -.

(24a) (24b) (25a)

(25b)

VOL. 59, 1968

PHYSICS: HARRISS AND HIRSCHFELDER

323

Indicial equation:

V0o2 VFo = f

(26a)

or

( ao°

= r cos 0

(26b)

Fo = r2 cos 0/2.

(26c)

= f ^- 02V . VFo V 602 -tF-

(27a)

(6F') = (r + 1) cos 0

(27b)

First iteration: or

F1 = r(1 + r/2) cos 0. (27c) Further iteration shows that F2 = FI so that we have the solution IoM = r(1 + r/2) cos 0 V/o. Evaluation of the second-order energy leads to =o(2= -2.25, which is the exact value and yields the correct value of 4.5 for the polarizability of the groundstate hydrogen atom. (B.2) Determination of #o(2) = G1o: #o(2) has two symmetry components,6 one transforming as Yoo(0,q), and one transforming as Y20(0,4b). The second-order perturbation equation can be solved treating G as a single function (in which case the procedure converges after four iterations) or the two components of G can be obtained separately. In what follows, the two components will be determined separately. The second-order perturbation equation (Ho - 0)Q,0(2) + (V -g0(l))0o(l) can be written, using earlier given values,

=

(2)(3)

-VFo -2.25#o, (28a) and operation on equation (28a) with the projection operators Aoo and A20 yields (28b) AooGj~o= (°°)GIwo

(Ho - Q)G4,o

AooVF4o

= - -(r2 + )0

A oooo =

A20oGo

A20VFJPO

=

(28d) (28e)

o

(2aO)GFo

(3 cos2 0 - 1) o (28f) 022g 3

= -

ao-w-= A20

(28c)

(11 +

2

(28g)

PHYSICS: HARRISS AND HIRSCHFELDER

324

PROC. N. A. S.

leaving the separate equations for (0o°)G and (2 0G /r2

(Ho - go)(0.O)GIP0 =

(Ho -&o)(2 0)GG

=

-

+

ra

+3 6 -

-

2.25 0°

(3 cos2 o

-

(28h)

1) V0.

(28i)

(B.2.a) Solution for (°0')G: Rewriting equation (28h) as V

.

(V2V (0 0)G)= -2

-

+

- 2.25) I/o2 =

(o)g

followed by the iterative sequence

Vqo2*V(oo)GO = (OO)g VF02 V(°)Gi+l = ((O2O)g

leads to the following set of (°0 )G1:

(0°O)Gi 0 + r3/9 + -2.25r 1 -2.251n r -2.25r + r2/3 + r3/4 + 2 2.25/r -2.251n r -1.25r + 3r2/4 + r3/4 + + 3r2/4 + r3/4 + 3 1.125/r -1.1251n r 4 3r2/4 + r3/4 + 5 3r2/4 + r3/4 +

r4/24 r4/24 r4/24 r4/24 r4/24 r4/24

or,

3r2/4 + r3/4 + r4/24. (B.2.b) Solution for (2. )G: Rewriting equation (28i) as (0 °)G =

V.* (o2V(2P)G)= - r (1 + r) (3 COS2

- 1)02

followed by the iterative sequence VO2- V (2 0?GO = (2,O)g V Ip02. V (2@ + = (2,O)g - 2V2(2 leads to (20 )Gi/(3 cos2 0 - 1) i 0 1 r2/6 + 2 5r2/16 + 3 5r2/16 +

r3/9 5r3/24 5r3/24 5r3/24

+ + + +

r4/24 r4/24 r4/24 r4/24

or,

(2.0)G = 5r2/16 + 5r3/24 + r4/24

(2,0) 9

VOL. 59P 1968

PHYSICS: HARRISS AND HIRSCHFELDER

325

and 0o(2) is thus determined to within an additive constant multiple of the zeroorder wavefunction, aAo. Evaluation of the normalization constant and collection of terms gives #o/(2) as

0o(2) = (° °)G/'o + (2 ')Gso + aco0 = 1/48{(36r2 + 12r3 + 2r4) +

(15r2 + 10r3 + 2r4)(3 cos2G

-

1) -

3721 i/o.

The fourth-order energy determined with this function is

go(4)= (o0(1) V - -0(g))0o(2){(#,o(2)J 4&o) + (4,o(1)J|o(1))} - 3555/64, which is the exact value, yielding the correct value of 10,665/8 as the hyperpolarizability of this system.7 * This work was supported by the National Aeronautics and Space Administration, grant NsG-275-62. t Permanent address: Department of Chemistry, University of Minnesota, Duluth. l Ovchinnikov, A. A., and A. D. Sukhanov, Sov. Phys. Dokl., 9, 685 (1965). 2 Dalgarno, A., and J. T. Lewis, Proc. Roy. Soc. (London), A233, 70 (1965). 3 Hirschfelder, J. O., W. B. Brown, and S. T. Epstein, Advances in Quantum Chemistry, ed P. 0. LUwdin, (New York: Academic Press, 1964), vol. 1, pp. 266, 271. 4Hylleraas, E. A., Z. Physik, 65, 209 (1930). 5 See, for example: Ince, E. L., Ordinary Differential Equations (New York: Dover Publications, Inc., 1956), pp. 47-49; or Webster, A. G., Partial Differential Equations of Mathematical Physics (New York: Dover Publications, Inc., 1966), chap. II. 6 This is easily shown through use of projection operators corresponding to particular rows of the irreducible representations of the group of HO. I Boyle, L. L., A. D. Buckingham, R. L. Disch, and D. A. Dunmur, J. Chem. Phys., 45, 1318 (1966).