Rayleigh-Ritz Method

solve(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 $S_{ij} = \langle \phi_{i} | \phi_{j} \rangle$.

solve(hamiltonian::Hamiltonian, basis::Basis; perturbation=Hamiltonian(), info=4)

This a solver for 1-basis calculations. This function returns solve(hamiltonian, BasisSet(basis); perturbation=perturbation, info=info).

solve(hamiltonian::Hamiltonian, basisset::GeometricBasisSet; perturbation=Hamiltonian(), info=4)

This function is a wrapper for solve(hamiltonian::Hamiltonian, basisset::BasisSet, ...).

function optimize(hamiltonian::Hamiltonian, basisset::BasisSet; perturbation=Hamiltonian(), info=4, progress=true, optimizer=Optim.NelderMead(), options...)

This function minimizes the energy by changing the exponents of the basis functions using Optim.jl.

\[\frac{\partial E}{\partial a_i} = 0\]

optimize(hamiltonian::Hamiltonian, basis::Basis; perturbation=Hamiltonian(), info=4, optimizer=Optim.NelderMead())

This a optimizer for 1-basis calculations. This function returns optimize(hamiltonian, BasisSet(basis); perturbation=perturbation, info=info, progress=progress, optimizer=optimizer, options...).

optimize(hamiltonian::Hamiltonian, basisset::GeometricBasisSet; perturbation=Hamiltonian(), info=4, optimizer=Optim.NelderMead())

This function minimizes the energy by optimizing $r_1$ and $r_n$ using Optim.jl.

\[\frac{\partial E}{\partial r_1} = \frac{\partial E}{\partial r_n} = 0\]

BasisSet(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:

\[\begin{aligned} \phi_1(r) &= \exp(-13.00773 ~r^2), \\ \phi_2(r) &= \exp(-1.962079 ~r^2), \\ \phi_3(r) &= \exp(-0.444529 ~r^2), \\ \phi_4(r) &= \exp(-0.1219492 ~r^2). \end{aligned}\]

BS = BasisSet(
  SimpleGaussianBasis(13.00773),
  SimpleGaussianBasis(1.962079),
  SimpleGaussianBasis(0.444529),
  SimpleGaussianBasis(0.1219492),
)

GeometricBasisSet(basistype, r₁, rₙ, n; nₘᵢₙ=1, nₘₐₓ=n)

This is a basis set with exponentials generated by geometric().

Exponents of Gaussian basis functions are given by geometric progression:

\[\begin{aligned} & v_i = \frac{1}{r_i^2}, \\ & r_i = r_1 a^{i-1}. \end{aligned}\]

This function return array of $\nu_i$:

\[(r_1, r_{n}, n, n_\mathrm{max}) \mapsto (\nu_1, \nu_2, \cdots, \nu_{n-1}, \nu_n, \nu_{n+1}, \cdots, \nu_{n_\mathrm{max}})\]

Usually $n = n_\mathrm{max}$. Set $n<n_\mathrm{max}$ if you want to extend the geometric progression.

Examples:

julia> ν = TwoBody.geometric(0.1, 10.0, 5)
5-element Vector{Float64}:
 100.0
  10.0
   0.9999999999999997
   0.09999999999999996
   0.009999999999999995

julia> ν = TwoBody.geometric(0.1, 10.0, 5, nₘₐₓ = 10)
10-element Vector{Float64}:
 100.0
  10.0
   0.9999999999999997
   0.09999999999999996
   0.009999999999999995
   0.0009999999999999994
   9.999999999999994e-5
   9.999999999999992e-6
   9.999999999999991e-7
   9.999999999999988e-8

SimpleGaussianBasis(a=1)

Position-Space

\[\phi_i(\pmb{r}) = \exp(-a_i r^2)\]

Momentum-Space

\[\phi_{i}(\pmb{k}) = \frac{1}{(2a_i)^{\frac{3}{2}}} \exp(-k^2/4a_i)\]

Proof (Fourier Transform)

\[\begin{aligned} \phi_{n}(\pmb{k}) &= \frac{1}{\sqrt{2 \pi}^3} \int \phi_{n}(\pmb{r}) \mathrm{e}^{\mathrm{i} \pmb{k} \cdot \pmb{r}} \mathrm{d}\pmb{r} \\ &= \frac{1}{\sqrt{2 \pi}^3} \int \phi_{n}(\pmb{r}) \mathrm{e}^{\mathrm{i} \pmb{k} \cdot \pmb{r}} r^2 \sin (\theta) ~\mathrm{d}r \mathrm{d}\theta \mathrm{d} \varphi \\ &= \frac{1}{\sqrt{2 \pi}^3} \iiint \mathrm{e}^{-\alpha_i r^2} \sqrt{4\pi} Y_{00}(\hat{\pmb{r}}) \left[ 4 \pi \sum_{l'=0}^{\infty} \sum_{m=-l'}^{l'} \mathrm{i}^{l'} j_{l'}(pr) Y_{l'm'}(\hat{\pmb{k}}) Y_{l'm'}^*(\hat{\pmb{r}}) \right] r^2 \sin\theta~ \mathrm{d} r \mathrm{d} \theta \mathrm{d} \varphi \\ &= \frac{1}{\sqrt{2 \pi}^3} 4 \pi \sqrt{4\pi} \sum_{l'=0}^{\infty} \sum_{m=-l'}^{l'} \left[ \mathrm{i}^{l'} Y_{l'm'}(\hat{\pmb{k}}) \int_0^{2 \pi} \int_0^\pi Y_{00}(\hat{\pmb{r}}) Y_{l'm'}^*(\hat{\pmb{r}}) \sin (\theta)~ \mathrm{d} \theta \mathrm{d} \varphi \int_0^{\infty} j_{l'}(pr) \mathrm{e}^{-\alpha_i r^2} r^{2} \mathrm{d}r \right]\\ &= \frac{1}{\sqrt{2 \pi}^3} 4 \pi \sqrt{4\pi} \sum_{l'=0}^{\infty} \sum_{m=-l'}^{l'} \left[ \mathrm{i}^{l'} Y_{l'm'}(\hat{\pmb{k}}) \delta_{0l'} \delta_{0m'} \int_0^{\infty} j_{l'}(kr) \mathrm{e}^{-\alpha_i r^2} r^{2} \mathrm{d}r \right] \\ &= \frac{1}{\sqrt{2 \pi}^3} 4 \pi \sqrt{4\pi} \mathrm{i}^{0} Y_{00}(\hat{\pmb{k}}) \int_0^{\infty} j_{0}(kr) \mathrm{e}^{-\alpha_i r^2} r^{2} \mathrm{d}r \\ &= \frac{1}{2\pi\sqrt{2\pi}} 4 \pi \frac{\sqrt{4\pi}}{\sqrt{4\pi}} \sqrt{\frac{\pi}{2}} \sqrt{\frac{2}{\pi}} \int_0^{\infty} j_{0}(kr) \mathrm{e}^{-\alpha_i r^2} r^{2} ~\mathrm{d}r \\ &= \frac{1}{(2\alpha_i)^{\frac{3}{2}}} \mathrm{e}^{-\frac{k^2}{4 \alpha_i}} \end{aligned}\]

Formula

plane-wave expansion in spherical harmonics:

\[\mathrm{e}^{\mathrm{i} \pmb{k} \cdot \pmb{r}} = 4 \pi \sum_{l=0}^{\infty} \sum_{m=-l}^{l} \mathrm{i}^{l} j_{l}(pr) Y_{lm}(\hat{\pmb{k}}) Y_{lm}^*(\hat{\pmb{r}})\]

special case of spherical harmonics:

\[Y_{00}(\hat{\pmb{r}}) = \frac{1}{\sqrt{4\pi}}\]

orthonormality of spherical harmonics:

\[\int_0^{2\pi} \int_0^\pi Y_{lm}(\hat{\pmb{r}})^* Y_{l'm'}(\hat{\pmb{r}}) \sin(\theta) ~ \mathrm{d} \theta \mathrm{d} \varphi = \delta_{ll'} \delta_{mm'}\]

citation needed:

\[\sqrt{\frac{2}{\pi}} \int r^{l} j_l(kr) \mathrm{e}^{-\alpha r^2} r^{2} \mathrm{d} r = \frac{1}{(2\alpha)^{l+\frac{3}{2}}} k^l e^{-\frac{k^2}{4\alpha}}\]

ContractedBasis([c1, c2, ...], [basis1, basis2, ...])

\[\phi' = \sum_i c_i \phi_i\]

matrix(basisset::BasisSet)

This function returns the overlap matrix $\pmb{S}$. The element is written as $S_{ij} = \langle \phi_{i} | \phi_{j} \rangle$.

matrix(hamiltonian::Hamiltonian, basisset::BasisSet)

This function returns the Hamiltonian matrix $\pmb{H}$. The element is written as $H_{ij} = \langle \phi_{i} | \hat{H} | \phi_{j} \rangle$.

element(SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} S_{ij} = \langle \phi_{i} | \phi_{j} \rangle &= \int \phi_{i}^*(r) \phi_{j}(r) \mathrm{d} \pmb{r} \\ &= \iiint \mathrm{e}^{-\alpha_i r^2} \mathrm{e}^{-\alpha_j r^2} ~r^2 \sin\theta ~ \mathrm{d} r \mathrm{d} \theta \mathrm{d} \varphi \\ &= \int_0^{2\pi} \mathrm{d}\varphi \int_0^\pi \sin\theta ~\mathrm{d}\theta \int_0^\infty r^{2} \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= 2\pi \times 2 \times \frac{1!!}{2^{2}} \sqrt{\frac{\pi}{a^{3}}} \\ &= \underline{\left( \frac{\pi}{\alpha_i + \alpha_j} \right)^{3/2}} \end{aligned}\]

Integral Formula:

\[\int_0^{\infty} r^{2n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}\]

element(o::Hamiltonian, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} H_{ij} &= \langle \phi_{i} | \hat{H} | \phi_{j} \rangle \\ &= \langle \phi_{i} | \sum_k \hat{o}_k | \phi_{j} \rangle \\ &= \sum_k \langle \phi_{i} | \hat{o}_k | \phi_{j} \rangle \\ \end{aligned}\]

element(o::RestEnergy, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} \langle \phi_{i} | mc^2 | \phi_{j} \rangle &= mc^2 \langle \phi_{i} | \phi_{j} \rangle \\ &= mc^2 \iiint \phi_{i}^*(r) \phi_{j}(r) ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= mc^2 \int_0^{2\pi} \mathrm{d}\varphi \int_0^\pi \sin\theta ~\mathrm{d}\theta \int_0^\infty r^{2} \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= mc^2 \times 2\pi \times 2 \times \frac{1!!}{2^{2}} \sqrt{\frac{\pi}{a^{3}}} \\ &= \underline{mc^2 \left( \frac{\pi}{\alpha_i + \alpha_j} \right)^{3/2}} \end{aligned}\]

Integral Formula:

\[\int_0^{\infty} r^{2n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}\]

element(o::Laplacian, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} \langle \phi_{i} | \nabla^2 | \phi_{j} \rangle = \underline{ -6 \frac{\alpha_i \alpha_j \pi^{\frac{3}{2}}}{(\alpha_i + \alpha_j)^{\frac{5}{2}}} } \end{aligned}\]

element(o::NonRelativisticKinetic, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

Derivation (without Green's identity)

\[\begin{aligned} T_{ij} = \langle \phi_{i} | \hat{T} | \phi_{j} \rangle &= \iiint \mathrm{e}^{-\alpha_i r^2} \left[ -\frac{\hbar^2}{2\mu} \nabla^2 \right] \mathrm{e}^{-\alpha_j r^2} ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= -\frac{\hbar^2}{2\mu} \iiint \mathrm{e}^{-\alpha_i r^2} \left[ \nabla^2 \right] \mathrm{e}^{-\alpha_j r^2} ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= -\frac{\hbar^2}{2\mu} \iiint \mathrm{e}^{-\alpha_i r^2} \left[ -6\alpha_j + 4\alpha_j^2 r^2 \right] \mathrm{e}^{-\alpha_j r^2} ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= -\frac{\hbar^2}{2\mu} \iint \sin\theta ~\mathrm{d}\theta \mathrm{d}\varphi \int \left[ -6\alpha_j + 4\alpha_j^2 r^2 \right] r^2 \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= -\frac{\hbar^2}{2\mu} \cdot 4\pi \left[ -6\alpha_j \mathrm{GGI}(2, \alpha_i + \alpha_j) +4\alpha_j^2 \mathrm{GGI}(4, \alpha_i + \alpha_j) \right] \\ &= -\frac{\hbar^2}{2\mu} \cdot 4\pi \left[ -6\alpha_j \frac{\Gamma\left( \frac{3}{2} \right)}{2 (\alpha_i + \alpha_j)^{\frac{3}{2}}} +4\alpha_j^2 \frac{\Gamma\left( \frac{5}{2} \right)}{2 (\alpha_i + \alpha_j)^{\frac{5}{2}}} \right] \\ &= -\frac{\hbar^2}{2\mu} \cdot 4\pi \left[ -6\alpha_j \frac{ \sqrt{\pi}/2}{2 (\alpha_i + \alpha_j)^{\frac{3}{2}}} +4\alpha_j^2 \frac{3\sqrt{\pi}/4}{2 (\alpha_i + \alpha_j)^{\frac{5}{2}}} \right] \\ &= -\frac{\hbar^2}{2\mu} \cdot 4\pi \left[ \frac{\alpha_j}{\alpha_i + \alpha_j} - 1 \right] \cdot 6 \alpha_j \cdot \frac{\sqrt{\pi}/2}{2 (\alpha_i + \alpha_j)^{\frac{3}{2}}} \\ &= -\frac{\hbar^2}{2\mu} \cdot 4\pi \left[ - \frac{\alpha_i}{\alpha_i + \alpha_j} \right] \cdot 6 \alpha_j \cdot \frac{\sqrt{\pi}/2}{2 (\alpha_i + \alpha_j)^{\frac{3}{2}}} \\ &= \underline{ \frac{\hbar^2}{2\mu} \cdot 6 \cdot \frac{\alpha_i \alpha_j \pi^{\frac{3}{2}}}{(\alpha_i + \alpha_j)^{\frac{5}{2}}} } \end{aligned}\]

Derivation (with Green's identity)

\[\begin{aligned} T_{ij} = \langle \phi_{i} | \hat{T} | \phi_{j} \rangle &= \iiint \mathrm{e}^{-\alpha_i r^2} \left[ -\frac{\hbar^2}{2\mu} \nabla^2 \right] \mathrm{e}^{-\alpha_j r^2} ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= -\frac{\hbar^2}{2\mu} \iiint \mathrm{e}^{-\alpha_i r^2} \nabla^2 \mathrm{e}^{-\alpha_j r^2} ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= \frac{\hbar^2}{2\mu} \iiint \left[ \nabla \mathrm{e}^{-\alpha_i r^2} \right] \left[ \nabla \mathrm{e}^{-\alpha_j r^2} \right] ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= \frac{\hbar^2}{2\mu} \iiint \left[ -2 \alpha_i r \mathrm{e}^{-\alpha_i r^2} \right] \left[ -2 \alpha_j r \mathrm{e}^{-\alpha_j r^2} \right] ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= \frac{\hbar^2}{2\mu} \cdot 4 \alpha_i \alpha_j \iiint \left[ r \mathrm{e}^{-\alpha_i r^2} \right] \left[ r \mathrm{e}^{-\alpha_j r^2} \right] ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= \frac{\hbar^2}{2\mu} \cdot 4 \alpha_i \alpha_j \iint \sin\theta ~\mathrm{d}\theta \mathrm{d}\varphi \int r^4 \mathrm{e}^{- (\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= \frac{\hbar^2}{2\mu} \cdot 4 \alpha_i \alpha_j \cdot 4 \pi \cdot \mathrm{GGI}(4, \alpha_i + \alpha_j) \\ &= \frac{\hbar^2}{2\mu} \cdot 4 \alpha_i \alpha_j \cdot 4 \pi \cdot \frac{\Gamma\left( \frac{5}{2} \right)}{2 (\alpha_i + \alpha_j)^{\frac{5}{2}}} \\ &= \frac{\hbar^2}{2\mu} \cdot 4 \alpha_i \alpha_j \cdot 4 \pi \cdot \frac{3\sqrt{\pi}/4}{2 (\alpha_i + \alpha_j)^{\frac{5}{2}}} \\ &= \underline{ \frac{\hbar^2}{2\mu} \cdot 6 \cdot \frac{\alpha_i \alpha_j \pi^{\frac{3}{2}}}{(\alpha_i + \alpha_j)^{\frac{5}{2}}} } \end{aligned}\]

Formula

Green's first identity:

\[\begin{aligned} \iiint_V f \pmb{\nabla}^2 g ~ \mathrm{d}V + \iiint_V \pmb{\nabla} f \cdot \pmb{\nabla} g ~ \mathrm{d}V = \iint_{\partial V} f \pmb{\nabla} g \cdot \pmb{n} ~ \mathrm{d}S \end{aligned}\]

generalized Gaussian integral:

\[\begin{aligned} \mathrm{GGI}(n,b) = \int_0^{\infty} x^{n} \exp \left(-b x^2\right) \mathrm{d}x = \frac{\Gamma\left( \frac{n+1}{2} \right)}{2 b^{\frac{n+1}{2}}} \end{aligned}\]

element(o::ConstantPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} \langle \phi_{i} | c | \phi_{j} \rangle &= c \langle \phi_{i} | \phi_{j} \rangle \\ &= c \iiint \phi_{i}^*(r) \phi_{j}(r) ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= c \int_0^{2\pi} \mathrm{d}\varphi \int_0^\pi \sin\theta ~\mathrm{d}\theta \int_0^\infty r^{2} \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= c \times 2\pi \times 2 \times \frac{1!!}{2^{2}} \sqrt{\frac{\pi}{a^{3}}} \\ &= \underline{c \left( \frac{\pi}{\alpha_i + \alpha_j} \right)^{3/2}} \end{aligned}\]

Integral Formula:

\[\int_0^{\infty} r^{2n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}\]

element(o::LinearPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} \langle \phi_{i} | r | \phi_{j} \rangle &= \iiint \phi_{i}^*(r) \times r \times \phi_{j}(r) ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= \int_0^{2\pi} \mathrm{d}\varphi \int_0^\pi \sin\theta ~\mathrm{d}\theta \int_0^\infty r^3 \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= 2\pi \times 2 \times \frac{1!}{2 (\alpha_i + \alpha_j)^{2}} \\ &= \underline{\frac{2\pi}{(\alpha_i + \alpha_j)^2}} \end{aligned}\]

Integral Formula:

\[\int_0^{\infty} r^{2n+1} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{n!}{2 a^{n+1}}\]

element(o::CoulombPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} \langle \phi_{i} | \frac{1}{r} | \phi_{j} \rangle &= \iiint \phi_{i}^*(r) \times \frac{1}{r} \times \phi_{j}(r) ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= \int_0^{2\pi} \mathrm{d}\varphi \int_0^\pi \sin\theta ~\mathrm{d}\theta \int_0^\infty r \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= 2\pi \times 2 \times \frac{0!}{2 (\alpha_i + \alpha_j)} \\ &= \underline{\frac{2\pi}{\alpha_i + \alpha_j}} \end{aligned}\]

Integral Formula:

\[\int_0^{\infty} r^{2n+1} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{n!}{2 a^{n+1}}\]

element(o::PowerLawPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} \langle \phi_{i} | r^n | \phi_{j} \rangle &= \iiint \phi_{i}^*(r) \times r^n \times \phi_{j}(r) ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= \int_0^{2\pi} \mathrm{d}\varphi \int_0^\pi \sin\theta ~\mathrm{d}\theta \int_0^\infty r^{n+2} \mathrm{e}^{-(\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= 2\pi \times 2 \times \frac{\Gamma\left( \frac{n+3}{2} \right)}{2 (\alpha_i + \alpha_j)^{\frac{n+3}{2}}} \\ &= \underline{2\pi\frac{\Gamma\left( \frac{n+3}{2} \right)}{(\alpha_i + \alpha_j)^{\frac{n+3}{2}}}} \end{aligned}\]

Integral Formula:

\[\int_0^{\infty} r^{n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{\Gamma\left( \frac{n+1}{2} \right)}{2 a^{\frac{n+1}{2}}}\]

element(o::GaussianPotential, SGB1::SimpleGaussianBasis, SGB2::SimpleGaussianBasis)

\[\begin{aligned} \langle \phi_{i} | \exp(-br^2) | \phi_{j} \rangle &= \iiint \phi_{i}^*(r) \times \exp(-br^2) \times \phi_{j}(r) ~r^2 \sin\theta ~\mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi \\ &= \int_0^{2\pi} \mathrm{d}\varphi \int_0^\pi \sin\theta ~\mathrm{d}\theta \int_0^\infty r^2 \mathrm{e}^{-(b+\alpha_i + \alpha_j) r^2} ~\mathrm{d}r \\ &= 2\pi \times 2 \times \frac{1!!}{2^{2}} \sqrt{\frac{\pi}{(b + \alpha_i + \alpha_j)^{2\cdot1+1}}} \\ &= \underline{\left( \frac{\pi}{b + \alpha_i + \alpha_j} \right)^{3/2}} \end{aligned}\]

Integral Formula:

\[\int_0^{\infty} r^{2n} \exp \left(-a r^2\right) ~\mathrm{d}r = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}}\]