Coulomb exponential profiles

A specified spatial benchmark

Use atomic units, $\hbar=m_e=e^2/(4\pi\epsilon_0)=1$, and ordinary three-dimensional coordinates. The auxiliary Coulomb Hamiltonian is

$$H_Z=\sum_{i=1}^2\left(-\frac12\Delta_i-\frac Z{r_i}\right) +\frac1{|x_1-x_2|}.$$

The two position labels describe two particles in the same spatial region. This benchmark supplies integration and variational tools. Applying it to V2's six-dimensional electron and actual Coulomb-plus-precession orbit requires a separately justified reduction and interaction law.

For $\zeta>0$, choose

$$\chi_\zeta(x)=\left(\frac{\zeta^3}{\pi}\right)^{1/2}e^{-\zeta r}, \qquad p_\zeta(r)=4\zeta^3r^2e^{-2\zeta r}.$$

The identity $\int_0^\infty r^ke^{-ar}\,dr=k!/a^{k+1}$ gives $\int p_\zeta=1$, $\langle r^{-1}\rangle=\zeta$, and, from $|\nabla\chi_\zeta|^2=\zeta^2|\chi_\zeta|^2$, kinetic energy $\zeta^2/2$ per particle.

Pair repulsion and optimized scale

The angular average of $1/|x-y|$ for radii $r,s$ is $1/\max(r,s)$: integrating $(r^2+s^2-2rsu)^{-1/2}/2$ over $u\in[-1,1]$ proves the assertion. Hence

$$\begin{aligned} J_\zeta &=\int_0^\infty\!\int_0^\infty \frac{p_\zeta(r)p_\zeta(s)}{\max(r,s)}\,dr\,ds\\ &=32\zeta^6\int_0^\infty r e^{-2\zeta r} \left(\int_0^r s^2e^{-2\zeta s}\,ds\right)dr =\frac{5\zeta}{8}. \end{aligned}$$

For example, substituting $\int_0^r s^2e^{-as}ds=[2-e^{-ar}(a^2r^2+2ar+2)]/a^3$ reduces the last integral to elementary factorial integrals.

The spatial product $\chi_\zeta(x_1)\chi_\zeta(x_2)$, multiplied by a normalized antisymmetric spin singlet, is an admissible fermion trial state for this Hamiltonian. Its energy is

$$E_Z(\zeta)=\zeta^2-2Z\zeta+\frac{5\zeta}{8} =\left(\zeta-Z+\frac5{16}\right)^2 -\left(Z-\frac5{16}\right)^2.$$

For $Z>5/16$, its minimum occurs at $\zeta=Z-5/16$. The variational principle gives $\inf\sigma(H_Z)\le -(Z-5/16)^2$ Hartree. For $Z=2$ the trial value is $-729/256$. This is an upper bound on the ground energy, with both the Hamiltonian and trial family specified.

A finite-range interaction bound

If a multiplication potential vanishes for $r>L$ and satisfies $|V(x)|\le V_0$, then

$$|\langle V\rangle|\le V_0\,\Pr(r\le L),\qquad \Pr(r\le L)=1-e^{-2\zeta L} [1+2\zeta L+2(\zeta L)^2] =\frac43(\zeta L)^3+O((\zeta L)^4).$$

This bounds a contact effect once its strength and range are supplied. More generally, a regular partial wave behaving as $r^\ell$ near zero has radial weight $O(r^{2\ell+2})$, so its probability within $L$ is $O(L^{2\ell+3})$. The exponent follows from the spatial measure; its coefficient depends on the normalized state.

• Gaussian contact profiles — complementary exact profile integrals.

• Actual orbit research correction — identifies the physical orbit target to which a reduction must connect.

• Resonance and profile tools — overview and verification.

resonance_profile_checks.py checks finite examples by quadrature, matrix evolution, and independent algebra.