Hamiltonian

Hamiltonian(operator1, operator2, ...)

\[\hat{H} = \sum_i \hat{o}_i\]

The Hamiltonian is the input for each solver. This is an example for the non-relativistic Hamiltonian of hydrogen atom in atomic units:

\[\hat{H} = - \frac{1}{2} \nabla^2 - \frac{1}{r}\]

H = Hamiltonian(
  Kinetic(hbar = 1, m = 1),
  Coulomb(coefficient = -1),
)

Kinetic(hbar=1, m=1)

\[-\frac{\hbar^2}{2m} \nabla^2\]

RestEnergy(c=1, m=1)

\[m c^2\]

Use c = 137.035999177 (from 2022 CODATA) in the atomic units.

RelativisticCorrection(c=1, m=1, n=2) The p^{2n} term of the Taylor expansion:

\[\begin{aligned} \sqrt{p^2 c^2 + m^2 c^4} =& m \times c^2 \\ &+ 1 / 2 / m \times p^2 (n=1) \\ &- 1 / 8 / m^3 / c^2 \times p^4 (n=2) \\ &+ 1 / 16 / m^5 / c^4 \times p^6 (n=3) \\ &- 5 / 128 / m^7 / c^6 \times p^8 (n=4) \\ &+ \cdots \end{aligned}\]

Use c = 137.035999177 (from 2022 CODATA) in the atomic units.

RelativisticKinetic(c=1, m=1)

\[\sqrt{p^2 c^2 + m^2 c^4} - m c^2\]

Use c = 137.035999177 (from 2022 CODATA) in the atomic units.

Constant(constant=1)

\[+ c\]

Linear(coefficient=1)

\[+ ar\]

Coulomb(coefficient=1)

\[+ \frac{a}{r}\]

PowerLaw(coefficient=1, exponent=1)

\[+ ar^n\]

Gaussian(coefficient=1, exponent=1)

\[+ a \exp(- b r^2)\]

Exponential(coefficient=1, exponent=1)

\[+ a \exp(- b r)\]

Yukawa(coefficient=1, exponent=1)

\[+ \frac{a}{r} \exp(- b r)\]

Delta(coefficient=1)

\[+ a δ(r)\]

Custom(f)

\[+ f(r)\]

Tabulated(R, V)