API reference
TwoBody.BasisTwoBody.BasisSetTwoBody.ConstantPotentialTwoBody.ContractedBasisTwoBody.CoulombPotentialTwoBody.DeltaPotentialTwoBody.ExponentialPotentialTwoBody.FunctionPotentialTwoBody.GaussianBasisTwoBody.GaussianPotentialTwoBody.HamiltonianTwoBody.LinearPotentialTwoBody.NonRelativisticKineticTwoBody.OperatorTwoBody.PowerLawPotentialTwoBody.RelativisticCorrectionTwoBody.RelativisticKineticTwoBody.RestEnergyTwoBody.SimpleGaussianBasisTwoBody.UniformGridPotentialTwoBody.YukawaPotentialTwoBody.elementTwoBody.elementTwoBody.elementTwoBody.elementTwoBody.elementTwoBody.elementTwoBody.solve
TwoBody.Basis — TypeBasis is an abstract type.
TwoBody.BasisSet — TypeBasisSet(basis1, basis2, ...)
\[\{ \phi_1, \phi_2, \phi_3, \cdots \}\]
The basis set is the input for Rayleigh–Ritz method. You can define the basis set like this:
\[\begin{aligned} +
API reference
TwoBody.BasisTwoBody.BasisSetTwoBody.ConstantPotentialTwoBody.ContractedBasisTwoBody.CoulombPotentialTwoBody.DeltaPotentialTwoBody.ExponentialPotentialTwoBody.FunctionPotentialTwoBody.GaussianBasisTwoBody.GaussianPotentialTwoBody.HamiltonianTwoBody.LinearPotentialTwoBody.NonRelativisticKineticTwoBody.OperatorTwoBody.PowerLawPotentialTwoBody.RelativisticCorrectionTwoBody.RelativisticKineticTwoBody.RestEnergyTwoBody.SimpleGaussianBasisTwoBody.UniformGridPotentialTwoBody.YukawaPotentialTwoBody.elementTwoBody.elementTwoBody.elementTwoBody.elementTwoBody.elementTwoBody.elementTwoBody.solve
TwoBody.Basis — TypeBasis is an abstract type.
TwoBody.BasisSet — TypeBasisSet(basis1, basis2, ...)
\[\{ \phi_1, \phi_2, \phi_3, \cdots \}\]
The basis set is the input for Rayleigh–Ritz method. You can define the basis set like this:
\[\begin{aligned} \phi_1(r) &= \exp(-13.00773 ~r^2), \\ \phi_2(r) &= \exp(-1.962079 ~r^2), \\ \phi_3(r) &= \exp(-0.444529 ~r^2), \\ @@ -9,12 +9,12 @@ SimpleGaussianBasis(1.962079), SimpleGaussianBasis(0.444529), SimpleGaussianBasis(0.1219492), -)
TwoBody.ConstantPotential — TypeConstantPotential(constant=1)
\[+ \mathrm{const.}\]
TwoBody.ContractedBasis — TypeContractedBasis([c1, c2, ...], [basis1, basis2, ...])
\[\phi' = \sum_i c_i \phi_i\]
TwoBody.CoulombPotential — TypeCoulombPotential(coefficient=1)
\[+ \mathrm{coeff.} \times \frac{1}{r}\]
TwoBody.DeltaPotential — TypeDeltaPotential(coefficient=1)
\[+ \mathrm{coeff.} \times δ(r)\]
TwoBody.ExponentialPotential — TypeExponentialPotential(coefficient=1, exponent=1)
\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r)\]
TwoBody.FunctionPotential — TypeFunctionPotential(f)
\[+ f(r)\]
TwoBody.GaussianBasis — TypeGaussianBasis(a=1, l=0, m=0)
\[\phi_i(r, θ, φ) = N _{il} r^l \exp(-a_i r^2) Y_l^m(θ, φ)\]
TwoBody.GaussianPotential — TypeGaussianPotential(coefficient=1, exponent=1)
\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r^2)\]
TwoBody.Hamiltonian — TypeHamiltonian(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} = +)
TwoBody.ConstantPotential — TypeConstantPotential(constant=1)
\[+ \mathrm{const.}\]
TwoBody.ContractedBasis — TypeContractedBasis([c1, c2, ...], [basis1, basis2, ...])
\[\phi' = \sum_i c_i \phi_i\]
TwoBody.CoulombPotential — TypeCoulombPotential(coefficient=1)
\[+ \mathrm{coeff.} \times \frac{1}{r}\]
TwoBody.DeltaPotential — TypeDeltaPotential(coefficient=1)
\[+ \mathrm{coeff.} \times δ(r)\]
TwoBody.ExponentialPotential — TypeExponentialPotential(coefficient=1, exponent=1)
\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r)\]
TwoBody.FunctionPotential — TypeFunctionPotential(f)
\[+ f(r)\]
TwoBody.GaussianBasis — TypeGaussianBasis(a=1, l=0, m=0)
\[\phi_i(r, θ, φ) = N _{il} r^l \exp(-a_i r^2) Y_l^m(θ, φ)\]
TwoBody.GaussianPotential — TypeGaussianPotential(coefficient=1, exponent=1)
\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r^2)\]
TwoBody.Hamiltonian — TypeHamiltonian(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:
\[\hat{H} = - \frac{1}{2} \nabla^2 - \frac{1}{r}\]
hamiltonian = Hamiltonian(
NonRelativisticKinetic(ℏ =1 , m = 1),
CoulombPotential(coefficient = -1),
-)TwoBody.LinearPotential — TypeLinearPotential(coefficient=1)
\[+ \mathrm{coeff.} \times r \]
TwoBody.NonRelativisticKinetic — TypeNonRelativisticKinetic(ℏ=1, m=1)
\[-\frac{\hbar^2}{2m} \nabla^2\]
TwoBody.Operator — TypeOperator is an abstract type.
TwoBody.PowerLawPotential — TypePowerLawPotential(coefficient=1, exponent=1)
\[+ \mathrm{coeff.} \times r^\mathrm{expon.}\]
TwoBody.RelativisticCorrection — TypeRelativisticCorrection(c=1, m=1, n=2) The p^{2n} term of the Taylor expansion:
\[\begin{aligned} +)
TwoBody.LinearPotential — TypeLinearPotential(coefficient=1)
\[+ \mathrm{coeff.} \times r \]
TwoBody.NonRelativisticKinetic — TypeNonRelativisticKinetic(ℏ=1, m=1)
\[-\frac{\hbar^2}{2m} \nabla^2\]
TwoBody.Operator — TypeOperator is an abstract type.
TwoBody.PowerLawPotential — TypePowerLawPotential(coefficient=1, exponent=1)
\[+ \mathrm{coeff.} \times r^\mathrm{expon.}\]
TwoBody.RelativisticCorrection — TypeRelativisticCorrection(c=1, m=1, n=2) The p^{2n} term of the Taylor expansion:
\[\begin{aligned} \sqrt{p^2 c^2 + m^2 c^4} =& m \times c^2 \\ &+ 1 / 2 / m \times p^2 (n=1) \\ @@ -22,7 +22,7 @@ &+ 1 / 16 / m^5 / c^4 \times p^6 (n=3) \\ &- 5 / 128 / m^7 / c^6 \times p^8 (n=4) \\ &+ \cdots -\end{aligned}\]
Use c = 137.035999177 (from 2022 CODATA) in the atomic units.
TwoBody.RelativisticKinetic — TypeRelativisticKinetic(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.
TwoBody.RestEnergy — TypeTwoBody.SimpleGaussianBasis — TypeSimpleGaussianBasis(a=1)
\[\phi_i(r) = \exp(-a_i r^2)\]
Note: This basis is not normalized. This is only for s-wave.
TwoBody.UniformGridPotential — TypeUniformGridPotential(R, V)
TwoBody.YukawaPotential — TypeYukawaPotential(coefficient=1, exponent=1)
\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r) / r\]
TwoBody.element — Methodelement(o::CoulombPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} +\end{aligned}\]
Use c = 137.035999177 (from 2022 CODATA) in the atomic units.
TwoBody.RelativisticKinetic — TypeRelativisticKinetic(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.
TwoBody.RestEnergy — TypeTwoBody.SimpleGaussianBasis — TypeSimpleGaussianBasis(a=1)
\[\phi_i(r) = \exp(-a_i r^2)\]
Note: This basis is not normalized. This is only for s-wave.
TwoBody.UniformGridPotential — TypeUniformGridPotential(R, V)
TwoBody.YukawaPotential — TypeYukawaPotential(coefficient=1, exponent=1)
\[+ \mathrm{coeff.} \times \exp(- \mathrm{expon.} \times r) / r\]
TwoBody.element — Methodelement(o::CoulombPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} \langle \phi_{i} | \frac{1}{r} | \phi_{j} \rangle &= \iiint \phi_{i}^*(r) @@ -36,13 +36,13 @@ &= \underline{\frac{2\pi}{\alpha_i + \alpha_j}} \end{aligned}\]
Integral Formula:
\[\begin{aligned} \int_0^{\infty} r^{2n+1} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{n!}{2 a^{n+1}} -\end{aligned}\]
TwoBody.element — Methodelement(o::Hamiltonian, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} +\end{aligned}\]
TwoBody.element — Methodelement(o::Hamiltonian, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} H_{ij} &= \langle \phi_{i} | \hat{H} | \phi_{j} \rangle \\ &= \langle \phi_{i} | \hat{T} + \hat{V} | \phi_{j} \rangle \\ &= \langle \phi_{i} | \hat{T} | \phi_{j} \rangle + \langle \phi_{i} | \hat{V} | \phi_{j} \rangle \\ &= T_{ij} + V_{ij} -\end{aligned}\]
TwoBody.element — Methodelement(o::LinearPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} +\end{aligned}\]
TwoBody.element — Methodelement(o::LinearPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} \langle \phi_{i} | r | \phi_{j} \rangle &= \iiint \phi_{i}^*(r) @@ -56,7 +56,7 @@ &= \underline{\frac{2\pi}{(\alpha_i + \alpha_j)^2}} \end{aligned}\]
Integral Formula:
\[\begin{aligned} \int_0^{\infty} r^{2n+1} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{n!}{2 a^{n+1}} -\end{aligned}\]
TwoBody.element — Methodelement(o::NonRelativisticKinetic, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} +\end{aligned}\]
TwoBody.element — Methodelement(o::NonRelativisticKinetic, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} T_{ij} = \langle \phi_{i} | \hat{T} | \phi_{j} \rangle &= \iiint \mathrm{e}^{-\alpha_i r^2} @@ -159,7 +159,7 @@ \cdot 6 \cdot \frac{\alpha_i \alpha_j \pi^{\frac{3}{2}}}{(\alpha_i + \alpha_j)^{\frac{5}{2}}} } -\end{aligned}\]
TwoBody.element — Methodelement(o::PowerLawPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} +\end{aligned}\]
TwoBody.element — Methodelement(o::PowerLawPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} \langle \phi_{i} | r^n | \phi_{j} \rangle &= \iiint \phi_{i}^*(r) @@ -171,7 +171,7 @@ \int_0^\infty r^{n+2} \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= 2\pi \times 2 \times \frac{\Gamma\left( \frac{n+3}{2} \right)}{2 (\alpha_i + \alpha_j)^{\frac{n+3}{2}}} \\ &= \underline{2\pi\frac{\Gamma\left( \frac{n+3}{2} \right)}{(\alpha_i + \alpha_j)^{\frac{n+3}{2}}}} -\end{aligned}\]
Integral Formula:
\[\int_0^{\infty} r^{n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{\Gamma\left( \frac{n+1}{2} \right)}{2 a^{\frac{n+1}{2}}}\]
TwoBody.element — Methodelement(SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} +\end{aligned}\]
Integral Formula:
\[\int_0^{\infty} r^{n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{\Gamma\left( \frac{n+1}{2} \right)}{2 a^{\frac{n+1}{2}}}\]
TwoBody.element — Methodelement(SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)
\[\begin{aligned} S_{ij} = \langle \phi_{i} | \phi_{j} \rangle &= \iiint @@ -183,4 +183,4 @@ \int_0^\infty r^{2} \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= 2\pi \times 2 \times \frac{1!!}{2^{2}} \sqrt{\frac{\pi}{a^{3}}} \\ &= \underline{\left( \frac{\pi}{\alpha_i + \alpha_j} \right)^{3/2}} -\end{aligned}\]
Integral Formula:
\[\int_0^{\infty} r^{2n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}\]
TwoBody.solve — Methodsolve(hamiltonian::Hamiltonian, basisset::BasisSet)
This function returns the eigenvalues $E$ and eigenvectors $\pmb{c}$ for
\[\pmb{H} \pmb{c} = E \pmb{S} \pmb{c}.\]
The Hamiltonian matrix is defined as $H_{ij} = \langle \phi_{i} | \hat{H} | \phi_{j} \rangle$. The overlap matrix is defined as $S_{ij} = \langle \phi_{i} | \phi_{j} \rangle$.
Settings
This document was generated with Documenter.jl version 1.7.0 on Saturday 9 November 2024. Using Julia version 1.11.1.
Integral Formula:
\[\int_0^{\infty} r^{2n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}\]
TwoBody.solve — Methodsolve(hamiltonian::Hamiltonian, basisset::BasisSet)
This function returns the eigenvalues $E$ and eigenvectors $\pmb{c}$ for
\[\pmb{H} \pmb{c} = E \pmb{S} \pmb{c}.\]
The Hamiltonian matrix is defined as $H_{ij} = \langle \phi_{i} | \hat{H} | \phi_{j} \rangle$. The overlap matrix is defined as $S_{ij} = \langle \phi_{i} | \phi_{j} \rangle$.