AUGUST 15, 1939 PHYSICAL REVIEW VOLUME 5 6

Forces in Molecules

R. P. Feynman

Massachusetts Institute oj Technology, Cambridge, Massachusetts

(Received June 22, 1939)

Formulas have been developed to calculate the forces in a molecular system directly, ratherthan indirectly through the agency of energy. This permits an independent calculation of theslope of the curves of energy vs. position of the nuclei, and may thus increase the accuracy, or

decrease the labor involved in the calculation of these curves. The force on a nucleus in an

atomic system is shown to be just the classical electrostatic force that would be exerted on thisnucleus by other nuclei and by the electrons' charge distribution. Qualitative implications ofthis are discussed.

MANY of the problems of molecular structure entire process is repeated for a new nuclearare concerned essentially with forces. The position, and the new value of energy calculated.

stiffness of valence bonds, the distortions in Proceeding in this way, a plot of energy vs.geometry due to the various repulsions and position is obtained. The force on a nucleus isattractions between atoms, the tendency of of course the slope of this curve.valence bonds to occur at certain definite angles The following method is one designed towith each other, are some examples of the kind obtain the forces at a given configuration, whenof problem in which the idea of force is para- only the configuration is known. It does notmount. require the calculations at neighboring configura-

Usually these problems have been considered tions. That is, it permits a calculation of thethrough the agency of energy, and its changes slope of the energy curve as well as its valye,with changing configuration of the molecule, for any particular configuration. It is to beThe reason for this indirect attack through emphasized that this allows a considerable savingenergy, rather than the more qualitatively illumi- of labor of calculations. To obtain force under thenating one, by considerations of force, is perhaps usual scheme the energy needs to be calculatedtwofold. First it is probably thought that force for two or more different and neighboring con-is a quantity that is not easily described or calcu- figurations. Each point requires the calculationlated by wave mechanics, while energy is, and of the wave functions for the entire system.second, the first molecular problem to be solved In this new method, only one configuration, theis the analysis of band spectra, strictly a problem one in question, need have its wave functionsof energy as such. It is the purpose of this paper computed in detail. Thus the labor is consider-to show that forces are almost as easy to calculate ably reduced. Because it permits one to get anas energies are, and that the equations are quite independent value of the slope of the energyas easy to interpret. In fact, all forces on atomic curve, the method might increase the accuracynuclei in a molecule can be considered as purely in the calculation of these curves, being especiallyclassical attractions involving Coulomb'

s law. helpful in locating the normal separation, orThe electron cloud distribution is prevented position of zero force.from collapsing by obeying Schrodinger>s equa- In the following it is to be understood that thetion. In these considerations the nuclei are nuclei of the atoms in the molecule, or otherconsidered as mass points held fixed in position, atomic system, are to be held fixed in position,

A usual method of calculating interatomic as point charges, and the force required to beforces runs somewhat as follows. applied to the nuclei to hold them is to be

For a given, fixed configuration of the nuclei, calculated. This will lead to two possible defini-the energy of the entire system (electrons and tions of force in the nonsteady state, for thennuclei) is calculated. This is done by the variation the energy is not a definite quantity, and themethod or other perturbation schemes. This slope of the energy curve shares this indefinite-

340

FORCES IN MOLECULES 341

ness. It will be shown that these two possibledefinitions are exactly equivalent in the steady-state case, and, of course, no ambiguity shouldarise there.

Let X be one of any number of parameterswhich specify nuclear positions. For example,X might be the x component of the position ofone of the nuclei. A force fx is to be associatedwith X in such a way that fxd\ measures thevirtual work done in displacing the nucleithrough d\. This will define the force only whenthe molecule is in a steady state, of energy 27,for then we can say f\ = - dU/d\. In the non-steadv-state case we have no sure guide to adefinition of force. For example, if U = f\p*H\pdvbe the average energy of the system of wavefunction ip and Hamiltonian H, we might define

fx'=-a(Z7)/ax'. (1)

Or again, we might take fx to be the average of- dH/dX or

/dH\ c dHfx=-( - )=-U*-fdv. (2)

\ax/A» J ax

We shall prove that under steady-state con-ditions, both these definitions of force becomeexactly equivalent, and equal to -dU/d\, theslope of the energy curve. Since (2) is simplerthan (1) we can define force by (2) in general.In particular, it gives a simple expression for theslope of the energy curve.

Thus we shall prove, when H\p = U\p andf \p\j/*dv - 1 that,

du r dH-= I 4**-i'dv.

ax J axNow

U= Jwhence,

dU r dH pdtp* f dip- = I i//*-ypdv-\- I-Hipdv-\- I ip*H-dv.

ax J ax J ax J ax

Since H is a self-adjoint operator,

/dip rd\p

\p*H-dv= I -H4>*dv.ax J ax

But H\p = U\p and Hyp* = U\p* so that we canwrite,

a U f dH r d\p* p d\p-= f ip*

~-\f/dv-\-U I ÿ--ipdv-\-U I -\p*dv.ax J ax J ax J ax

These last two terms cancel each other sincetheir sum is,

V- f *P*iPdv= Z7-(1) = 0.ax J ax

Whence

dU r dH- = I -ipdvax J ax

in the steady state. This much is true, regardlessof the nature of H, (whether for spin, or nuclearforces, etc.). In the special case of atomic systemswhen H=T+V where T is the kinetic energyoperator, and V the potential, since dH/d\= a F/aX we can write

dU r dVfx =f\=-= - | -dv. (3)

ax J ax

The actual calculation of forces in a real

molecule by means of this theorem is not im-practical. The J

~

ip*4'(dV/d'

\)dv is not too differ-ent from J

~

Tp*\pVdv, which must be calculated ifthe energy is to be found at all in the variationalmethod. Although the theorem (3) is the mostpractical for actual calculations, it can bemodified to get a clearer qualitative picture ofwhat it means. Suppose, for example, the systemfor which \p is the wave function contains severalnuclei, and let the coordinates of one of thesenuclei, a, be X°, Y", Z" or X»a where 1, 2, 3,mean X, Y, Z. If we take our X parameter to beone of these coordinates, the resultant force onthe nucleus a in the fi direction will be givendirectly by

/„° = - J*

W*{d V/dXf)dv

from (3).Now V is made up of three parts, the inter-

action of all nuclei with each other (Vap), of eachnucleus with an electron (Fp,-), and of each

342 R. P. FEYNMAN

electron with every other (Vi,) ; or

F=E Vap+Z Fÿ+E Vif.", 0 0. i i, i

Suppose x/ are the coordinates of electron i,and as before XF° those of nucleus a of charge qa.Then V$i - qÿe/Rÿ, where

-?ÿ2=x;(*/-v)J.M-l

So we see that

dVpi dV/3, dVn-=- - and that -= 0.

dX„S dx/ dX„t

Then (3) leads to

r d Vai f d Va$}"=+ I WE---dv-Y,

I -H>*dvJ i dx/ 0 J dXp

"

r dVair pp ~

\ dVa0= E-: i H*dv \dv-E-~ (4)

J » " J 0 dXp

X

since dVai/dx/ does not involve any electroncoordinate except those of electron i. fif- ÿ -dvmeans the integral over the coordinates of allelectrons except those of electron i. The lastterm has been reduced since 6 Vÿ/dxÿ" does notinvolve the electron coordinates, and is constantas far as integration over these coordinates goes.This term gives ordinary Coulomb electrostaticrepulsion between the nuclei and need not beconsidered further. Now efiSfty*dv is just thecharge density distribution pi(x) due to electron i,where e is the charge on one electron. Theelectric field Eÿix*) at any point due to thenucleus a is {\/e)dVai/dxvi, so that (4) may bewritten

r d VaB

J i 0 dX„

«

The 3N space for N electrons has been reducedto a 3 space. This can be done since £,«(**)depends only on x' and is the same function of x'no matter which i we pick. This implies thefollowing conclusion:

The force on any nucleus (considered fixed) inany system of nuclei and electrons is just theclassical electrostatic attraction exerted on the

nucleus in question by the other nuclei and by

the electron charge density distribution for allelectrons,

p(x) = Ep.-O)-t

It is possible to simplify this still further.Suppose we construct an electric field vector Fsuch that

V-F= -47rpO); VXF= 0.

Now from the derivation of Ewe know that it

arises from the charge qa on nucleus a, so thatV-E0 = 0 except at the charge a where its integralequals qa. Further,

d Va0- E-=2«[E-E/]at x".

ÿ s 31/ 0

Then

1 r dVaS/„ÿ=-(V ÿ F)Ell"dv - E-

471-*' 0

1 r d Va0= H-I Fu{V-Ef)dv-Y,

-

AwJ 0 6x„"

- qa{.

FlT\a,\,xe'JrqJi'£,E/~]&ix" (5)

0

the transformation of the integral being accom-plished by integrating by parts. Or finally, theforce on a nucleus is the charge on that nucleustimes the electric field there due to all the

electrons, plus the fields from the other nuclei.This field is calculated classically from thecharge distribution of each electron and fromthe nuclei.

It now becomes quite clear why the strongestand most important attractive forces arise whenthere is a concentration of charge between twonuclei. The nuclei on each side of the concen-

trated charge are each strongly attracted to it.Thus they are, in effect, attracted toward eachother. In a Hj molecule, for example, the anti-symmetrical wave function, because it must bezero exactly between the two H atoms, cannotconcentrate charge between them. The sym-metrical solution, however, can easily permitcharge concentration between the nuclei, andhence it is only the solution which is sym-metrical that leads to strong attraction, and theformation of a molecule, as is well known. It is

SOURCE OF INTERNAL FRICTION 343

clearly seen that concentrations of charge be-tween atoms lead to strong attractive forces, andhence, are properly called valence bonds.

Van der Waals' forces can also be interpretedas arising from charge distributions with higherconcentration between the nuclei. The Schrod-

inger perturbation theory for two interactingatoms at a separation R, large compared to theradii of the atoms, leads to the result that thecharge distribution of each is distorted fromcentral symmetry, a dipole moment of order1/i?' being induced in each atom. The negative

charge distribution of each atom has its center ofgravity moved slightly toward the other. It isnot the interaction of these dipoles which leadsto van der Waals' force, but rather the attractionof each nucleus for the distorted charge dis-tribution of its own electrons that gives theattractive l/R7 force.

The author wishes to express his gratitude toProfessor J. C. Slater who, by his advice andhelpful suggestions, aided greatly in this work.He would also like to thank Dr. W. C. Herringfor the latter's excellent criticisms.