Variational Energy Lowering May Increase Hamiltonian Dispersion Paul Blaise Universite de Valenciennes et du Hainaut Cambresis, I.FO.RE.P, 59326 Valenciennes cedex, France Olivier Henri-Rousseau Universite de Perpignan, 66025 Perpignan cedex. France Generally i t is very difficult to solve the time-independent Schrodinger equation, but i t is always possible to get approximate solution either by perturbation or variational methods (I).While the perturbation method may give energies that are less than the experimental one, this is not so for the variational method, which has the advantage of leading to energies that approach the experimental value from above when the number of variational parameters is increased. Moreover, a judicious choice of variational parameters may allow the system to verify someimportant theorems of quantum mechanics such as the virial and the Hellmann-Feynman theorems (2). I t is then assumed that the wavefunction has been improved by the variational procedure. Our purpose is to show on the example of the molecular ion HZ+that this assumption may become questionable if one considers as a criterion of accuracy the dispersion of the Hamiltonian of the system. The Hamlltonlan of the Molecular Ion HZ+(2) Let us look at the hydrogen molecular ion Hzf in the framework of the Born-Oppenheimer approximation for which the single electron is considered as moving in the skeleton of the two fixed protons. The Hamiltonian H of the system is
respectively, and V,,. is the nucleus-nucleus repulsion operator in au.,
where R is the distance between the two fixed protons (see Fig. 1). The total exact energy of the system inits fundamental state is the lowest eigenvalue E" of the Hamiltonian H. Because T and Vdo not commute with H, the kinetic and the potential energies are average values when the system is in an eigenstate III.) of H. Virial Theorem and Hellmann-Feynman Theorem for H ~ + Now let us look a t the virial(2) and the Hellmann-Feynman (3) theorems. For a system such as a diatomic molecule the quantum (2) virial theorem leads to: ($4q+)= - E " + R ddl3 R (6)
where T, the kinetic operator of the electron, and V the potential operator are respectively given in atomic units (au) by
T = -%V2
(2)
v = v,. + v,
(3)
Here V , is the Coulomhic electron-nuclei attraction operator, in au,
r, and rz are the distances of the electron to protons 1and 2,
Figure 1. The hydrogen molecular ion H2+system.
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Number 1 January 1988
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whereas the Hellmann-Feynman theorem states that the force acting on the nuclei is equal to the average value of the partial derivative of the Hamiltonian with respect to the internuclear distance.
In what follows we shall focus our attention on the equilihrit the internuclear distance. um distance R.0 with r e s ~ e c to that is,
For this special situation the virial theorem gives
whereas the Hellmann-Feynman theorem gives 1
where V , is given above, it being understood that the kinetic energy operator Tdoes not depend on R. I t must be kept in mind that in eqs 10 and 11the average values are performed on the I#) which is an eigenket of the Hamiltonian. When we have in place of I#) the approximate ket lq), i t may be shoivn that the virial and the HellmannFeynman theorems hold if special variational procedures are used. This leads, in place of eqs 10 and 11, to and
with r the radial spherical coordinate of the electron. (oO is then the crude LCAO wavefunction that we shall denote @'A.Now i t is well known that the virial theorem is verified if theu parameter is determined hv the variational procedure. On theother hand it has heen shown ( 4 ) that the HellmannFeynman theorem holds if the floating parameter x is optimized by the variation method. Such considerations allow one to identifv. in addition to the crude LCAO wavefunction @A,which ides not obey the above theorems, the three following functions: pa,, where x = 0 and a is optimized, which verifies only the virial theorem (5). @c. where x is optimized and 0 = 1, which obeys only the Hellmann-Feynman theorem (6),and v0owhere x are optimized and for which hoth theorems hold (6). Since there are two optimized variational parameters in p " ~ , whereas there is only one in @ B and p0c and none in PA, it may be inferred from the variational Ritz theorem that the average value of the Hamiltonian will he lowered when we pass from @ Ato &or e 0 c and from V'B or qoc to POD. The question arises whether the accuracy of these wavefunctions is increased in the same way. Criterion of the Accuracy of a Wavefunction Before answering the question of the accuracy of the wavefunction, it is important to define clearly the somewhat vague term ''accuracy". Besides the variational method, which may he considered as a test of accuracy for a wavefunction, it is possible to define accuracy by the behavior of the local energy E = H+l+, which must be the same everywhere in space for an eigenfunction ( I ) . Linked to that there is the possibility of defining more rigorously the increase in accuracy by verifying for all integers n that the differences (H") - (H)"
Some Trial Wavefunctions for Hz+ Let us look a t an approximate ket Ipo) of the lowest "' eigenket of H, I+o). In the {lr)Jrepresentation, I @ becomes @(r) = (+Po).
We shall consider for @(r) trial wavefunctions involving flexible generality in order to satisfy, if necessary, the virial (2) or the Hellmann-Feynman (4) theorems. For this purpose we shall look a t the function, where a is a variational parameter and A a normalization constant, whereas r ' ~and r'z are related to rl and rz through simple relations that can he deduced from Figure 2. If a = 1and x = 0, we may recognize (2) in eq 15 a linear combination of the degenerate hydrogen 1s atomic orbitals centered on the two protons, the expressions of which are of the form,
(15)
are vanishing. As a matter of fact, for a central force potential V, the greater will he the n moment of the Hamiltonian and the smaller will be the distance t o the nuclei which will ~ l a the v fundamental role throueh r-". As a conseouence. l: mainly concern &on; hi& values of the moment of H w of large potential and thus large local kinetic energy of the electron. This shows that the accuracy defined in those terms (see eq 15) is not without link with a definition of the accuracy in terms of the cusp behavior (7), which concerns the origins of the pocentials (position of the fixed nuclei). In a first approximation the accuracy, in terms of eq 15, may he approached 11). considering only the first moment of H that is, by assuming that the dispersion of the Hamiltonian, A H = l j m
tends to zero when the accuracy increases. I t may he ohserved that the dispersion on H is also used in general variation (8) method which gives hoth an upper and lower limit for energy levels Er through the inequality, (H)+AHtE,>(H)-AH
(16)
Accuracy of the Hz+ %,+ Wavefunctlons in Terms of AH Now let us look a t the accuracy of the wavefundions of the above section in terms of the smallness of the dispersion on the Hamiltonian. The dispersion on H is AH = d(9°1flv0) - (9°w190)2
(17)
From eq 1we have for the second moment of H W=F+V+TV+VT
(18)
Moreover, since the trial wavefunctions are real, Figure 2. T M floating parameter x.
10
Journal of Chemical Education
(v0ITVa")= (9'lVllv0)
(19)
Mean Values (au) Calculated on Several Trial Wavetunctlons for the Fundamental State of Hz+ Function
VA PB VC VD
R.
(T)
2.500 2.000
0.3827
(v)
(H)
AH
(HF)
2.443
-0.9476 -0.5648 0.5864 -1.1729 -0.5865 0.3834 -0.9545 -0.5711
0.1607 -0.0547 -0.0868 0.5178 -0.00005
2.000
0.6020 -1.2046 -0.6026
0.0124
0,5942
-0,5941
0.1998
0,7125 ~ 0 , 0 0 0 1 3
0.00143
then the calculation of AH involves only the average values of p, and TV,These mean values lead to unusual inte. grals that may be calculated by a numerical procedure using Gauss+Legendre method (9),These values are given in the tahle with that of the quantity (HF) defined by
v,
which would he zero if lvO)were 1110) (see eq 1) or satisfied the variational Hurley condition ( 4 ) (see above). Examhation of the table shows that the average value of the Hamiltonian is lowered when we pass from VA to w~ and vc and when we nass -from ran or m . "r to mn , - in aereement with the Ritz theorem, and that the Hellmann-~&man (HF) theorem is ~
A~~~~~
~~
.-
~
However, it must he noted that there are very accurate wavefunctions, such as that of Rothstein (21,which satisfy both the virial and the Hellman-Fevnman theorems for which &is much smaller than that oitained in the case of VA (see last row of the table). I t is important to note that the dispersion on H is more increased in order to satisfy the Hellmann-Feynman theorem than it is in order to satisfy the virial theorem. The reason is that the introduction of floating orbitals in order to satisfy the Hellmann-Feynman leads to the appearance of two new singularities in the energy at the origin of the floating orbitals. It is necessary to have in mind that, for usual approximate wavefunctions, there are two singularities in the local kinetic and potential energies that are linked to the position of the nuclei. When floating orbitals are considered, there is the apparition of two new singularities at the origins of these orbitals because the singularities in the kinetic energy are translated in these positions, whereas the singularities in potential energy remain a t the position of the nuclei. The above results show that the constraints reauired to decrease the second moment of the Hamiltonian are not necessarilv the same as those that are rewired to satish, bv variationai procedure, the virial and thk Hellmann-F&iman theorems. They are an illustration of the fact that lowering the energy by variational method is not a proof hut only a presumption in favor of the accuracy improvement of the-trial wavefunctions. Literature Cited ~-
~~
point out that the dispersion on the Hamilcnnian increases a,hen passing from on to VH or 6(:and from $11 or vc to GI,, contrarv ro what would he expected for the accuracv of the
~~
R ~ h not. ~ isa.56, s i d 4. H ~ r 1 c y . AC I hr Ho% SLC i ~ n dn, 19S4..42%. 179 i. ~ ~ k ~ N~ . ; H~ O ~t O ~ ~cUi E ~P m3a,r , ~ I.9 Z a . a I M 6 t b ~ , ~I). I I .shdt 11. J J rhrm rh>*.19% 29,866 b.,.
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