× ∫ ∞ 0 n ∏ j =1 ( e β$_{j}$$^{ε$_{j}$}$dε$_{j}$ ) δ ( ε$_{0}$ − B$_{0}$ − E$_{c}$$_{0}$ − n ∑ j =1 ( p 2 j 2 m$_{j}$ + ε$_{j}$ − B$_{j}$ − E$_{cj}$ ) . (A4) ω$_{n}$ = k ∏ l =1 1 N$_{l}$ ! n ∏ j =1 ( V$_{n}$ (2 π ℏ ) 3 ) n − 1 ∫ n ∏ j =1 d $^{3}$p$_{j}$ δ ( n ∑ j =1 ⃗ p$_{j}$ ) ( 1 2 πi ) n n ∏ j $_{=1}$( ∫ c$_{j}$ + i ∞ c$_{j}$ − i ∞ dβ$_{j}$ e − β$_{j}$f ∗ $_{j}$( β$_{j}$ ) ) × exp [ β$_{1}$ ( ε$_{0}$ − B$_{0}$ − E$_{c}$$_{0}$ − n ∑ j =1 ( p 2 j 2 m$_{j}$ − B$_{j}$ − E$_{cj}$ ) )] n ∏ j =2 ∫ ∞ 0 e ( β$_{j}$ − β 1 ) $^{ε$_{j}$}$dε$_{j}$ . (A5) ω$_{n}$ = k ∏ l =1 1 N$_{l}$ ! n ∏ j =1 ( V$_{n}$ (2 π ℏ ) 3 ) n − 1 ∫ n ∏ j =1 d $^{3}$p$_{j}$ δ ( n ∑ j =1 ⃗ p$_{j}$ ) ( 1 2 πi ) n n ∏ j $_{=1}$( ∫ c$_{j}$ + i ∞ c$_{j}$ − i ∞ dβ$_{j}$ e − β$_{j}$f ∗ $_{j}$( β$_{j}$ ) ) × exp [ β$_{1}$ ( ε$_{0}$ − B$_{0}$ − E$_{c}$$_{0}$ − n ∑ j =1 ( p 2 j 2 m$_{j}$ − B$_{j}$ − E$_{cj}$ ) )] n ∏ j =2 1 β$_{1}$ − β$_{j}$ . (A6) − 1 2 πi ∫ c$_{j}$ + i ∞ c$_{j}$ − i ∞ dβ$_{j}$ e − β$_{j}$f ∗ $_{j}$( β$_{j}$ ) β$_{j}$ − β$_{1}$ = e − β 1 f ∗ $_{j}$( β 1 ) , (A7) ω$_{n}$ = k ∏ l =1 1 N$_{l}$ ! ( V$_{n}$ (2 π ℏ ) 3 ) n − 1 ∫ n ∏ j =1 d $^{3}$p$_{j}$ δ ( n ∑ j =1 ⃗ p$_{j}$ ) (A8) × 1 2 πi ∫ c + i ∞ c − i ∞ dβ exp [ β ( ε$_{0}$ − B$_{0}$ − E$_{c}$$_{0}$ − n ∑ j =1 ( p 2 j 2 m$_{j}$ + f ∗ j ( β ) − B$_{j}$ − E$_{cj}$ ) )] . ∫ n ∏ j =1 d $^{3}$p$_{j}$ δ ( n ∑ j =1 ⃗ p$_{j}$ ) exp [ − β n ∑ j =1 p 2 j 2 m$_{j}$ ] = 1 (2 π ) 3 ∫ d $^{3}$x n ∏ j =1 ∫ d $^{3}$p$_{j}$ exp [ i⃗ x · n ∑ j =1 ⃗ p$_{j}$ − β n ∑ j =1 p 2 j 2 m$_{j}$ ] 15 We next perform the integrals over the momenta. Using the integral representation of the function, we have since ℜ [ β$_{1}$ ] > ℜ [ β$_{j}$ ] . After performing the β$_{j}$ integrals, j = 2 , . . . , n , we have (taking β$_{1}$ → β ) We next perform the integrals over β$_{j}$ , j = 2 , . . . , n . To do this, we use the fact that the function exp [ − β$_{j}$f ∗ j ( β$_{j}$ ) ] is well behaved (no singularities) for ℜ [ β$_{j}$ ] > 0 . In particular, we note that as ℜ [ β$_{j}$ ] → ∞ , exp [ − β$_{j}$f ∗ j ( β$_{j}$ ) ] → g$_{j}$ , where g$_{j}$ is the ground-state degeneracy of fragment j . We can thus close the β$_{j}$ contour to the right, to obtain The remaining integrals over ε$_{2}$$_{,}$ . . . , ε$_{n}$ can now be performed if we take ℜ [ β$_{1}$ ] > ℜ [ β$_{j}$ ] , j = 2 , . . . , n . The result is We begin by integrating over the excitation energies. The first integral, over ε$_{1}$ , for example, furnishes