Calculations on Some Simple Systems Using a New Variation Principle BY G. G. HALL Dept. of Mathematics The University Nottingham Received 1 1 th October 1968 Calculations of upper and lower bounds for the energies of some one-electron atoms and other systems are discussed. These employ a new form of variation principle which permits the use of discontinuous trial wavefunctions and which is generally superior to the traditional principle. The implications of this principle for future calculations are also discussed. 1. INTRODUCTION In practical applications traditional forms of variation principle such as the Rayleigh-Ritz principle suffer from a number of disadvantages. There are various circumstances for example in which it would be advantageous to use trial functions which are discontinuous.This is not possible because the form of the functional requires the trial function to be differentiable. The variational principle described here uses a functional which does not involve differentiation and so allows for trial functions which are discontinuous. If the trial function is differentiable then this principle leads to lower upper bounds to the energy than the older principle. This improved accuracy is achieved at the expense of a doubling of the dimensions of the integrals which have to be evaluated. The evaluation of these integrals for many-electron systems has proved to be difficult and the examples quoted here are of one-electron systems. 2. A MATRIX INEQUALITY The basic principle behind this variation principle can be illustrated most simply by reference to a similar principle for finite matrices.If A is a positive definite Hermitean matrix the Rayleigh-Ritz principle asserts that for any trial vector u the ratio satisfies where A. is also asserts p = U~AU/U~U (1) > & 7 (2) the smallest eigenvalue. that the ratio Since A-1 is also Hermitean the Rayleigh-Ritz v = u~A-'u/u~u (3) is less than the largest eigenvalue of A-l which is just A; so that v-l > 2 0 . (4) For given u A-lu can be found by solving a set of linear equations and consequently v-l can be calculated readily. 69 70 NEW VARIATION PRINCIPLE These two upper bounds for Lo can be related using the Cauchy-Schwarz in- equality for the positive definite matrix A which states that and consequently Thus the additional effort in calculating v-l is repaid by a more accurate estimate of Lo.This can be illustrated using the simple matrix (utAu)(utA-'u) > (u~u)~ (5) p>v-? 5 3 - 1 * = (-; -; -:> For the trial vector (O,O,l) and for the more accurate vector (0,1,3) 213 (8) p = 1 ; v-1 = (9) p = 315 = 0.6; v-' = 417 = 0.57. Since ib0 = 0.55 the improved estimates obtained from v - I are significantly better than those from p. 3. VARIATION PRINCIPLE The new form of variation principle was introduced in a letter and established formally in a subsequent paper '. The Schrodinger equation for the system which is where T is the kinetic energy operator and V the potential is first generalized by introducing a new parameter E and writing (T+V)+r = Er$r (10) (T+ ~ / E V ) X = AX or Green's operator G is the operator reciprocal to (A-T) VX = &(A - T)x.G(A) = ( A - T)-l (1 3) GVX = E X . (14) where 1. is now a fixed negative number so that eqn. (12) becomes This operator can always be found in the form of an integral operator over a known kernel. This equation in which E appears as the eigenvalue of GV with ;1 fixed is known as the conjugate eigenvalue equation. For boundstates of Coulombic systems both G and V are negative definite so that the E eigenvalues are all positive. Since the product of two Hermitean operators is not generally Hermitean the conjugate eigenvalue equation is not in convenient form until it is made Hermitean by operating with V Thus V on the right of (15) acts as a weighting function in the Hilbert space of the eigenfunctions. The Rayleigh-Ritz principle for eqn. ( 1 5) is the variation principle now proposed.It asserts that the ratio VGVX = EVX. (15) G . G. HALL 71 for any trial function co such that the integral Jw*Vcod.r converges is always less than the largest eigenvalue E~ W E O . If the system is Coulombic eqn. (1 1) can be immediately reduced to the original Schrodinger equation by means of a scale factor such that It follows that the ground state energy Eo satisfies the bound In this expression A may be varied to minimize the bound and this is the equivalent of introducing a scale factor into co and optimizing it. The bound given by eqn. (19) can be compared with that given by the usual ratio. As before the use of the reciprocal operator means that a better result is always obtained. ~ r 2 = Erin; xr(x) = $ r ( x / & r ) * (18) Eo < Lq2. (19) 4. ILLUSTRATIVE C A L C U LA T I 0 N S Since the kinetic energy term does not appear in (16) the ratio q can be given a meaning even for a discontinuous trial function.The only restriction is that co should belong to the function space whose basis vectors are the eigenfunctions of the conjugate eigenvalue equation. This will be so provided that the integral sco* Vcodz (20) is convergent. be inadmissible we may consider the trial function As an illustration of the use of the principle for a function which would otherwise for the hydrogen atom. In this function a is a parameter and the lowest energy is -0.2905 obtained when a = 2.483. Thus despite the discontinuity at r = a and the lack of a cusp at r = 0 about 60 % of the exact energy is found. Much better energies are obtained using more realistic functions even if they still contain dis- continuities.The elimination of the kinetic energy operator has another consequence for the theory. Several of the best known lower bound formulae depend on a calculation of the variance of the energy. For the Schrodinger equation this is not often feasible since the term involving the square of the kinetic energy operator may not be defined or may diverge. For the conjugate eigenvalue equation this difficulty does not arise and in principle the variance can always be calculated although in practice it may be difficult to evaluate all the integrals. The variance for a trial function w becomes c2 = fw*VGVGVcodz/Jw*Vwdz-q2. (22) The Weinstein lower bound is then Eo > A(q + To illustrate this lower bound formulae we consider electron moving in a one-dimensional potential box of function is (23) the simple example of an length 1 Bohr.The tria 1 inside 0 outside 72 NEW VARIATION PRINCIPLE and the zero of potential is adjusted so that the potential inside is 1 Hartree. In this example with the potential everywhere positive the bounds are reversed and the variation principle gives a lower bound to the ground state energy of 574 Hartrees. The Weinstein relation gives an upper bound of 6.29 Hartrees. The exact energy is 5.935 Hartrees. 5. CALCULATION OF MEAN VALUES Many atomic and molecular properties depend essentially on the mean value of some operator. If the ground state wavefunction is not known accurately those mean values can be considerably in error. These errors can be progressively elimi- nated by a repeated use of the conjugate eigenvalue equation.Since the ground state corresponds to the largest eigenvalue of GV the effect of GV on a trial function is to increase the weight of the ground state eigenfunction in the result. This filtering operation can be repeated if necessary until sufficiently accurate results are obtained. The effect is exactly comparable with the power method of finding the largest eigenvalue of a matrix. If the operator in question is F the simplest mean value is Jo*FcodT/i co*codT. jw*FG Vcodz / Jw*GVwdT One stage of filtering gives while two stages gives the more symmetrical result 6. PROSPECTS The illustrations used above are of a simple kind so as to display the advantages of this new formulation. The prime motive for using discontinuous trial functions is that they enable more elaborate systems such as molecules and solids to be considered in termsof their localized parts. If these parts correspond to trial functions which vanish outside non-overlapping cells then a considerable reduction can be made in the number of integrals that require evaluation. Several molecular examples are now being considered. In the molecular as in the atomic examples the evaluation of the integrals has presented difficulties. These are due primarily to their high dimensionality and relative unfamiliarity. The possibilities of Gaussian methods of integration are now being investigated. It may be that these difficulties will prevent the method being applied more generally than to the integral equation analogue of the Hartree- Fock equations. G. G. Hall Chem. Physics Letters 1967 1 495. G. G. Hall J. Hyslop and D. Rees Znt. J. Quantum. Chem. in press.