Gaussian contact profiles

Spatial model

Consider two normalized densities on $m$ shared ordinary spatial coordinates of the nested physical manifold. A Gaussian is a specified local flat-space profile, with centre $X$ and positive-definite covariance matrix $\Sigma$:

$$\rho_{\Sigma,X}(x)= \frac{\exp[-\frac12(x-X)^T\Sigma^{-1}(x-X)]} {(2\pi)^{m/2}\sqrt{\det\Sigma}}, \qquad \int_{\mathbb R^m}\rho_{\Sigma,X}(x)\,d^mx=1.$$

Thus $\langle x\rangle=X$ and $\langle(x-X)(x-X)^T\rangle=\Sigma$. Covariance eigenvalues have units of length squared. The Gaussian describes spatial extent; an actual orbit or a time-averaged exposure requires its own dynamical specification.

For a local density contact energy, define

$$U(R)=g I(R),\qquad I(R)=\int\rho_{\Sigma_A,X_A}(x)\rho_{\Sigma_B,X_B}(x)\,d^mx, \qquad R=X_A-X_B.$$

The model supplies $g$. If $U$ has energy units, $g$ has units of energy times spatial volume. Representation contractions and other nonlocal kernels can be included in a larger model; the calculation here evaluates this particular scalar contact.

Exact convolution

Let $C=\Sigma_A+\Sigma_B$. Then

$$\boxed{ I(R)=\frac{\exp[-\frac12R^TC^{-1}R]} {(2\pi)^{m/2}\sqrt{\det C}}. }$$

For a direct derivation, use the Fourier convention $\widehat\rho(k)=\int e^{-ik\cdot x}\rho(x)\,d^mx$. Completing the Gaussian square gives $\widehat\rho_{\Sigma,X}(k)=e^{-ik\cdot X-\frac12k^T\Sigma k}$. The product integral is therefore

$$I(R)=\frac{1}{(2\pi)^m} \int_{\mathbb R^m}e^{-ik\cdot R-\frac12k^TCk}\,d^mk.$$

Diagonalize the real symmetric positive matrix $C$ by an orthogonal rotation. Each one-dimensional integral contributes $\sqrt{2\pi/c_j}\exp[-R_j^2/(2c_j)]$, where $c_j>0$ is an eigenvalue. Multiplication proves the formula. In particular, correlations and unequal widths are retained in $C$.

The relative contact profile and the absolute overlap are

$$\Omega(R)=\frac{I(R)}{I(0)} =e^{-\frac12R^TC^{-1}R}, \qquad I(0)=\frac{1}{(2\pi)^{m/2}\sqrt{\det C}}.$$

$\Omega$ is dimensionless; $I$ has inverse-volume units. Replacing $I$ by $\Omega$ in an energy absorbs the width-dependent $I(0)$ into its coupling. That replacement needs an explicit normalization convention.

Forces and widths

With the covariances held fixed during a displacement, set $W=C^{-1}$. The force on centre $A$, the opposite force on $B$, and the energy Hessian are

$$F_A=-\nabla_R U=gI(R)WR,\qquad F_B=-F_A,$$

$$\nabla_R\nabla_R U =gI(R)\left[WRR^TW-W\right].$$

Positive $g$ gives repulsion in this contact model; negative $g$ gives attraction. If a profile deforms with $R$, total derivatives additionally differentiate its covariance and normalization.

In the second row the wavefunction is proportional to $e^{-\sqrt{\lambda}|x-X|^2/2}$, so each density has $\sigma^2=1/(2\sqrt{\lambda})$. The overlap's Gaussian standard deviation is $L=\lambda^{-1/4}$ along each coordinate. An absolute physical length requires the scale and coefficients of the specified operator. The oscillator domain supplies its normalization and decay conditions.

Contact across nested depths

For a higher-depth density on coordinates $z\in\mathbb R^d$, choose an orthogonal shared-coordinate projection $x=Bz$, with $BB^T=\mathbf1_m$. Completing $B$ to an orthogonal coordinate frame identifies the remaining ordinary spatial coordinates $y$. A Gaussian with mean $X_B$ and covariance $\Sigma_B$ has full marginal

$$\bar\rho_B(x)=\int\rho_B(x,y)\,d^{d-m}y =\rho_{B\Sigma_BB^T,\,BX_B}(x).$$

Indeed its characteristic function is the higher one evaluated at $B^Tk$, which proves the covariance rule even when shared and remaining coordinates are correlated. Apply the contact formula with $C=\Sigma_A+B\Sigma_BB^T$ and $R=X_A-BX_B$.

The cylindrical contact premise uses a lower-depth density independent of $y$. Fubini then gives $\int\rho_A(x)\rho_B(x,y)\,dx\,dy=\int\rho_A(x)\bar\rho_B(x)\,dx$. This uses a cylindrical field normalized on its supported submanifold. A box-normalized probability extension instead carries its explicit inverse fibre-volume factor.

Every coordinate in this construction belongs to the same nested physical manifold. The projection describes the smaller sector's shared directions; the containing particle's dimensions remain equivalent in physical kind.

Verification and related

spatial_profile_checks.py checks direct density quadrature, anisotropic rotations, force and Hessian derivatives, and correlated Gaussian marginals.

• Interaction operator tools — navigation to the interaction and projection calculations.