Exchange splitting of the interaction energy and the multipole expansion of the wave function

Abstract

The exchange splitting J of the interaction energy of the hydrogen atom with a proton is calculated using the conventional surface-integral formula J surf [ϕ], the volume-integral formula of the symmetry-adapted perturbation theory J SAPT [ϕ], and a variational volume-integral formula J var [ϕ]. The calculations are based on the multipole expansion of the wave function ϕ, which is divergent for any internuclear distance R. Nevertheless, the resulting approximations to the leading coefficient j 0 in the large-R asymptotic series J(R) = 2e −R−1 R(j 0 + j 1 R −1 + j 2 R −2 + · · · ) converge, with the rate corresponding to the convergence radii equal to 4, 2, and 1 when the J var [ϕ], J surf [ϕ], and J SAPT [ϕ] formulas are used, respectively. Additionally, we observe that also the higher j k coefficients are predicted correctly when the multipole expansion is used in the J var [ϕ] and J surf [ϕ] formulas. The SAPT formula J SAPT [ϕ] predicts correctly only the first two coefficients, j 0 and j 1 , gives a wrong value of j 2 , and diverges for higher j n . Since the variational volume-integral formula can be easily generalized to many-electron systems and evaluated with standard basis-set techniques of quantum chemistry, it provides an alternative for the determination of the exchange splitting and the exchange contribution of the interaction potential in general.

Convergence properties of the multipole expansion of the exchange contribution to the interaction energy

“…In other words, the value of the asymptotically leading term of J var [Φ] depends only on the values of Φ on the line joining the nuclei a and b, where P l (cos θ a ) = P l ′ (cos θ b ) = 1, similarly as it was observed for J SAPT [Φ] in Ref. [36]. Thus, in the analysis of the convergence properties of approximations to j 0 it is sufficient to calculate the values of the functions f (k) n on the line joining the nuclei,…”

“…, and J var [Φ] with respect to the order of multipole expansion used to approximate the primitive wave function Φ were presented in our previous work [36]. By analyzing these numerical results we have found that the convergence rate of the leading term, j 0 , of the asymptotic expansion of Eq.…”

“…(28) of Ref. [36] is defined via φ n = f n φ 0 ; this definition has been erroneously omitted in the line preceding Eq. (28) …”

“…In this communication we shall present a rigorous mathematical derivation of the discovered convergence rates and the corresponding convergence radii (with respect to the order of multipole expansion of Φ) for j 0 calculated from the surface J surf [Φ] and variational J var [Φ] expression (in the case of J SAPT [Φ] such analysis has already been presented in Ref. [36]). Furthermore we derive closed-form formulas for partial sums of the series approximating j 0 and show that these partial sums converge to the correct value j 0 = −1.…”

“…(6) is expanded in powers of V , one obtains the SRS expansion of the exchange energy through the Kth order. In our recent work [36] we showed that the exchange energy can be very accurately calculated from the variational principle based formula J var [Φ]…”