The second virial coefficient is calculated by a new method which exploits certain results known from formal scattering theory. In particular it is shown that the essential quantity may be expressed as the determinant of the S matrix at a given energy. The present approach suggests several approximation techniques and also seems applicable to many-body problems

Goldberger, M. L.

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THE PHYSICS OF FLUIDS VOLUME 2, NUMBER 3 MAY-JUNE, 1959 
On the Second Virial Coefficient 
M. L. GOLDBERGER 
Palmer Physical Laboratory, Princeton University, Princeton, New Jersey 
(Received February 9, 1959) 
The second virial coefficient is calculated by a new method which exploits certain results known 
from formal scattering theory. In particular it is shown that the essential quantity may be expressed 
as the determinant of the S matrix at a given energy. The present approach suggests several approxi-
mation techniques and also seems applicable to many-body problems. 
T HERE is not the slightest doubt in anyone's mind as to how the second virial coefficient 
of a gas is calculated in quantum theory! The only 
excuse for giving yet another calculation is that the 
method is a powerful one and may perhaps be of 
use in other problems. 
What must be evaluated is the difference between 
the partition function for a two-particle system with 
and without interaction. We write this as 
where E,. and E,. are, respectively, the eigenvalues 
of the Hamiltonian with and without the interaction 
between the particles. Let us agree to ignore the 
trivial contribution from the center-of-mass motion 
and thus regard E,. and E,. as energies of the relative 
motion. We rewrite Eq. (1) as follows: 
n = ~ J. dse-~r[ L --1 - - 1-] 
27rl r. .. r - E,. r - E,. 
= 2;i f. dte-~r :r q: ln (t - E,.) 
- I: ln (s - E,.) 1 
(2) 
where the contour C extends from + ro above the 
real axis beyond the least of the eigenvalues E,., 
E,. and returns to + ro below the real axis.2 
Next we remark that ?r,.(s- E,.) is just the secular 
equation for the determination of the eigenvalues 
of the Schroedinger equation. It may, therefore, be 
written quite generally as det (s - H) where H = 
Ho + V is the full Hamiltonian and det A means 
the determinant of the operator A. Similarly 
'IT,.(t - E,.) = det (s - Ho). In order that these 
I See, for example, D. ter Haar, Elements of Statistical 
Mechanics (Reinhart and Company, Inc., New York, 1954), 
p. 193. 
252 
equations have meaning we must imagine that the 
system is enclosed in a box so that there are a finite 
number of discrete levels. The passage to the limit 
of infinite volume is entirely painless in the method 
to be developed here. 
We are led evidently to the study of a function 
D(s) defined by 
det (s - H) ( 1 ) 
D(s) = det (s - Ho) = det 1 - s - Ho V . (3) 
It is perhaps worth remarking that although D(s) 
does not actually exist because of the singular 
nature of the operator (s - H 0)-
1 V in the limit of 
a large volume and a continuous spectrum, our 
formal manipulations of it will lead to no error. As 
a matter of fact, only the imaginary part of In 
D(s = E + iTJ) (TJ -t 0) will be required for our 
purposes, and this does exist. For simplicity let us 
assume that the potential Vis spherically symmetric. 
We may discuss then each angular momentum 
separately and write D(r) = ll[D1(r}]
21 +\ since each 
energy is evidently (2l + 1) degenerate. 
The properties of D(s) are well known and are 
discussed in detail by Schwinger,3 DeWitt/ and 
Baker. 5 For ease of reference we derive in a very 
crude way the results of these authors which we 
need. The treatment of DeWitt is the most con-
venient for our purposes. We note first from Eq. (2) 
that we will ultimately require Dz (t) for s = E ± iTJ, 
i.e., infinitesimally near the real energy axis, so 
we confine our attention to D 1(E + iTJ) = D 1+(E); 
'This method was used by G. Wentzel [Helv. Phys. Acta 
15, 111 (1942)] in field theory. It is perhaps worth noting that 
the solution to the problem treated by Wentzel, just as in the 
calculation of virial coefficients, is most easily obtained by 
evaluating the change in the density of states produced by 
the interaction (see reference 1). Some techniques closely 
related to ones used in our work have been used in statistical 
mechanics problems by K. M. Watson [Phys. Rev. 103, 489 
(1956)]. 
3 J. Schwinger, Phys. Rev. 94, 1362 (1954) . 
. ~B. S. DeWitt, Phys. Rev. 103, 1565 (1956). Much of 
what we shall describe is completely covered in DeWitt's 
paper and is included here only for the benefit of those who 
are not experts in formal scattering theory. 
6 M. Baker, Ann. Phys. (N. Y.) 4, 271 (1958). 
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SECOND VIRIAL COEFFICIENT 253 
clearly D 1(E - iTt) = [D1+(E)]*. It follows from 
the fact that the number of energy levels of H and 
H 0 are the same that D 1+(± oo) = 1 and that as 
V ---+ 0, D 1+ ---+ 1. Further, as E increases from 
minus infinity, where the phase of D 1+(E) is zero, 
'when it crosses over the first bound state of H, the 
phase decreases by '~~"i if there are bound states of 
H 0 (which we assume are absent) the phase increases 
by r. Evidently this phase decrease of 1r happens 
at each bound state and if these occur for energies 
EB <o ( < 0), then we have 
Im ln DI+(E) = - L 'll"fJ(E- E 8 <1>)8(-E) 
B (4) 
- fJ(E)f,(E), 
where It( E) is yet to be determined, and fJ(x) is 
the step function, unity for x > 0, and zero for 
X< 0. 
To determine ft(E) we go back to the original 
expression for D [see Eq. (2)]. We have then for 
E > 0 (writing E+ = E + iTt, and E .. , = En~ + 
tlE,.1), 
TI (E+- E,.,) 
-ft(E) = Im ln TI (E+ _ E,.,) 
.. 
= lm ln n (1 - E ~"1 ) (5) 
n + Enl 
I "" tlE,., ~-m.L. . 
- .. E+- E,.j 
Next we must relate the energy shift E,.1 of the 
nth level to the spacing of the unperturbed spectrum. 
This is done by remarking that in the absence of 
interaction demanding that the radial wave function 
of the relative motion vanish on the surface of a 
large sphere of radius R determines the allowed 
wave numbers k,.0 according to k,.0 R - (lr/2) = n1r 
and thus ilE .. 1 = ... k .. 
0 /(m,R) is the spacing between, 
say, the nth and (n + 1)st level (m. is the reduced 
mass of the two-particle system). With interaction 
we have the wave numbers related to the phase 
shift of the asymptotic wave function 81 by k .. R -
(lr)/2 + 81 = n'll". Then 
tlE,., ,...., k,. o(k,. - k,.o) = k,.o 8.(E..,) 
m. -m.~ 
- - 6,(E,.,) AE 
- L.l nl• 
7r 
Evidently in the limit 1.\S R ---+ oo , ilE .. 1 ---+ 0 and the 
sum over n in Eq. (5) becomes an integral and we 
have in this same limit 
f1CB) = - lm 1"" dEE 8'~E) = 6t(E). (6) 
7r 0 + 'tfl - E 
Let us write now the result of the preceding con-
siderations as Im ln D 1+(E) = - tl 1(E) with 
tl,(E) = 7r L: fJ(E- EB(I))fJ( -E) + fJ(E)6t(E), (7) 
B 
where in order that tl 1 (E) be continuous at B = 0, 
we note that 81(0) = N< 0 '11" where N< 1> is the total 
number of bound states of angular momentum l. 
We note that 
(8) 
which has a very suggestive form: for E > 0, this 
is just the S matrix describing the scattering at 
energy E in the particular angular momentum 
state. We shall return to this point later. 
The evaluation of Eq. (2) may now be effected 
trivially. We write 
11 = ___!_f."" dEe- 11z !!:_ ln [D+(E)]* 
2ri EL dE D+(E) (Q) 
= _! !."" dEe_11z !!:_ Im ln D+(E), 
'II" EL dB 
where EL( < 0) is less than any of the bound state 
energies. By substituting D = II[Dt]21 + 1 and using 
Eq. (7) we have 
1J = 1_ f."" dBe-/IB L (2l + 1) 
'II" EL I 
·[r L 6(E- BB<n) + fJ(E) d 8'(E)J 
B dE (10) 
= L: L: (2l + 1) exp [ - {3 E s < n] 
B I 
+ ! 1"" dEe-fiB L (2l + 1) d ME) , 
'II" 0 I dE 
which is the correct answer. 1 It is important to note 
that if one integrates by parts in Eq. (10), the fact 
that 8(0) = N< 1>1r must be taken into account. 
These results may be generalized readily to the 
case where the potential is not spherically symmetric, 
so that the separation into angular moments is not 
possible. One needs simply replace the phase shifts 
61(E) by the eigenphase shifts of the S matrix, 
6A(E). These are the phase shifts in the states which 
diagonalize the S matrix. We write now 
[D+(E)]* = exp 2i{ L L 1rfJ(E- EB <A>) 
D+(E) ). B (11) 
· 8( -B) + fJ(E) L 6).(E)}. 
). 
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254 M. L. GOLDBERGER 
If we confine our attention from now on to E > 0 
we see that 
[~:f1:r = exp 2i ~ ~.(E) = det S(E), (12) 
where the identification made with the determinant 
of the S matrix at energy E is shown in detail by 
DeWitt.4 It is also evident from the fact that in its 
eigen-representation 
S(E) = L jX, E > e2if>x(E) < X, El. (13) 
X 
The expression of D + * / D + in terms of det S 
seems to be quite useful in connection with various 
approximation methods. Following DeWitt we write 
1 
E+ _ Ho (EL -H) = 1 - Go+(E)V, 
More precisely, it is defined as 
K(E) = V l - Jo(E)V 
= V + G0(E) VK(E), 
and in terms of this quantity 
L(E) = 1 - i1r~(E - H 0)K(E). 
1 + i1r~(E - H 0)K(b) 
(19) 
(20) 
We note also that the relevant (i.e., on the energy 
shells) matrix elements of K may be expressed in 
terms of the standing wave solution of the Schroe-
dinger equation 1/;k 01 (r) as 
~(Ek' - Ek)Kk'k = o(E'k' - Ek) 
(21) 
Go+(E) = E+ ~ Ho = E !: Ho - i7r~(E - Ho) and 
(14) 
where the P symbol means that in any integrations 
one takes the principle value. Introducing Go-
Go+*, form a quantity L(E) 
L(E) = [1 - G0 _(E) V] 1 _ G~+(E) V (15) 
It is evident by its construction that 
detL(E) = D+*(E)/D+(E), (16) 
and presumably then det L(E) = det S(E). Next 
we write 
1 
L(E) = 1 + (Go+ - Go-) V 1 _ a: V (17) 
= 1 - 21rio(E - Ho) V 1- ~--;+ V 
and it is from this form that DeWitt shows det 
L(E) = det S(E). It is important to note that the 
matrix elements of L(E) are not equal to those of 
the S matrix; only the determinants are equal. We 
proceed in a different manner and remark that 
1 
~(E - H o) V 1 - Go V 
. L(E) = 1 - 21ri . 1 · (18) 
1 + 'lr't~(E Ho)V 1 _ Go V 
This may be easily verified by multiplying numerator 
and denominator by 1 + Go V from the right. The 
combination of operators which occurs in (18) is 
known in scattering theory as the reaction matrix.6 
s See, for example, B. A. Lippmann and J. Schwinger, 
Phys. Rev. 79, 469 (1950). 
(22) 
where Ek = k 2 /2m,. In the case of a spherically 
symmetric potential we have6 (with lk'l = lkl = k) 
Kk'k - 2!:k .2: (2/ + 1) tan Mk)Pt(k' ·k). (23) 
An approximation which naturally suggests itself 
is an expansion of det L(E) in powers of K. This 
may be easily done using the formula det A 
exp {trace In A}. We have then immediately 
det L(E) = d et S(E) 
::::::::: exp {-trace 27ri~(E - H 0)K(E)} 
(24) 
{ J d
3
k' ( k'
2 
k
2 
) = exp - 21ri -:~ ~ - - -
(27r) 2m, 2m, 
· J d3re-•k' ''V(r)if;k <0 (r)} , 
(25) 
where Ek has been written as e /2m,. Using Eqs. 
(21) and (23) we have 
det S(E) "'exp {2i L (2l + 1) tan ME)}. (26) 
l 
The next term in the expansion involves (tan ~)3 • 
The first term in the expansion of ln det S(E) in 
powers of the interaction is trivially obtained from 
Eq. (25) by the replacement of if;k 01 (r) by exp 
(i k•r), in which case we have, carrying out the 
integration 
det S(E) ,.._, exp {- 2ik ~ J d3r V(r)} , (27) 
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SECOND VIRIAL COEFFICIENT 255 
and we note that - (m,/211") J d3rV(r) is just the 
Born approxim~tion to the forward scattering 
amplitude. The expansion may, of course, be 
obtained directly from Eq. (15) as 
det L(E) = det S(E) 
""' exp {- trace Go- V + trace Go+ V) 
= exp trace {[- E _ i: _ Ho (ZS) 
+ E + i:- nJv} 
= exp trace {- 211"io(E - H 0) V), 
which then leads directly to Eq. (27). 
We plan to apply the methods described in this 
paper to certain many-body problems. One possible 
virtue of this approach is that it concentrates 
directly on what is needed, namely the phase of 
D +(E), and never calculates anything irrelevant. 
Furthermore, it seems to avoid a difficulty found in 
several other approximation techniques7 which 
show an alarming tendency to yield as a presumably 
exact result in the low-density region something 
equivalent to Eq. (26). Methods have been devised 
to avoid this but they are rather involved.8 
7 K. A. Brueckner, Phys. Rev. 97, 1353 (1955); N. Rugen-
holtz (unpublished lecture, Princeton, New Jersey, December, 
1958). 
8 W. Riesenfeld and K. M. Watson, Phys. Rev. 104, 492 
(1956). 


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On the Second Virial Coefficient

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On the Second Virial Coefficient

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