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Hessian Matrix
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$$\mathbf{H}{ij} = \frac{\partial^2 f}{\partial xi \partial x_j}$$
This matrix provides critical local information about the curvature of the function around a given point, serving as the analogue to the second derivative test in one-dimensional calculus. In optimization theory, the nature of the Hessian matrix at a critical point (where the gradient is zero) determines whether that … -
Hessian Matrix
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Gradient Dampening Context
When calculating derivatives of quantum mechanical expectation values, one may encounter the phenomenon of Gradient Dampening. This unusual stability in the derivative of the density matrix is often observed when the system's orbital set is overly saturated, leading to an artificially high degree of conditioning in the Hessian matrix of the auxiliary functional. While standard [optimization algori… -
Minimum
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In multivariable calculus, local minima are frequently found by examining critical points\—where the gradient vector $\nabla f$ is zero, or undefined. For a differentiable function, the first derivative test indicates a necessary condition for a local minimum:
$$ \nabla f(c) = \mathbf{0} $$
To distinguish between a local minimum, maximum (mathematics)/), and saddle point, … -
Stationary Point In Chemistry
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Classification via Curvature Analysis
The true chemical nature of a stationary point is determined by the second derivative test-(via Curvature Analysis), which involves analyzing the Hessian matrix) of the potential energy function evaluated at that point. The Hessian matrix), composed of the second partial derivatives of the energy with respect to the nuclear coordinates, provides information about the curvature of the PES/) a…