Duy-Khoi Dang
Zimmerman Group
\[\begin{equation*} \hat{H} \vert \psi \rangle = E \vert \psi \rangle \end{equation*}\]
\[\begin{gather*} \fragment{1}{\hat{H}} \fragment{2}{= \color{lime}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{lime}{-\frac{1}{2} \sum_i \nabla_i^2}}_{\text{e$^-$ KE}}}} \fragment{3}{ \color{cyan}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{cyan}{-\frac{1}{2} \sum_I \nabla_I^2}}_{\text{nuc KE}}}} \fragment{4}{ \color{violet}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{violet}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert} }}_{\text{e$^-$-e$^-$ repulsion}}}} \fragment{5}{ \color{orange}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{orange}{+ \sum_{I>J} \frac{Z_I Z_J}{\vert R_I - R_J \vert} }}_{\text{nuc-nuc repulsion}}}} \fragment{6}{ \color{yellow}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{yellow}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}}_{\text{nuc-e$^-$ attraction}}}} \end{gather*}\]
\[\begin{gather*} \hat{H} = \color{lime}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{lime}{-\frac{1}{2} \sum_i \nabla_i^2}}_{\text{e$^-$ KE}}} \color{cyan}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\boxed{\color{cyan}{-\frac{1}{2} \sum_I \nabla_I^2}}}_{\text{nuc KE}}} \color{violet}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{violet}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert} }}_{\text{e$^-$-e$^-$ repulsion}}} \color{orange}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\boxed{\color{orange}{+ \sum_{I>J} \frac{Z_I Z_J}{\vert R_I - R_J \vert} }}}_{\text{nuc-nuc repulsion}}} \color{yellow}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{yellow}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}}_{\text{nuc-e$^-$ attraction}}} \end{gather*}\]
\[\begin{equation*} \hat{H} = \color{lime}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{lime}{-\frac{1}{2} \sum_i \nabla_i^2}}_{\text{e$^-$ KE}}} \color{violet}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{violet}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert} }}_{\text{e$^-$-e$^-$ repulsion}}} \color{yellow}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{yellow}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}}_{\text{nuc-e$^-$ attraction}}} \end{equation*}\]
$$\hat{H} \vert \psi \rangle = E \vert \psi \rangle$$ Only solvable for H-atom
\[\begin{equation*} \langle \psi \vert \hat{H} \vert \psi \rangle \geq E_0 \end{equation*}\]
Guess Wave function $\implies$ Optimize
Naive approach \[\begin{gather*} \fragment{2}{\psi=\phi_1(\mathbf{r}_1)\phi_2(\mathbf{r}_2)\cdots\phi_N(\mathbf{r}_N)} \fragment{3}{\impliedby \text{No exchange}} \\\\ \fragment{4}{\psi = \frac{1}{\sqrt{N!}} \begin{vmatrix} \phi_1(\mathbf{r}_1) & \phi_2 (\mathbf{r}_1) & \cdots & \phi_N(\mathbf{r}_1) \\ \phi_1(\mathbf{r}_2) & \phi_2 (\mathbf{r}_2) & \cdots & \phi_N(\mathbf{r}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_1(\mathbf{r}_N) & \phi_2 (\mathbf{r}_N) & \cdots & \phi_N(\mathbf{r}_N) \end{vmatrix} } \fragment{5}{\impliedby \text{has exchange}}\\\\ \fragment{6}{\implies\text{Optimize orbitals}} \end{gather*}\]
\[\begin{gather*} \fragment{4}{\hat{H}\psi = E \psi} \fragment{5}{\implies \mathbf{H}\mathbf{C} = E\mathbf{C}} \\ \fragment{6}{ H_{ij} = \langle \phi_i \vert \hat{H} \vert \phi_j \rangle} \end{gather*}\]
\[\begin{gather*} {\hat{H}\psi = E \psi} {\implies \mathbf{H}\mathbf{C} = E\mathbf{C}} \\ { H_{ij} = \langle \phi_i \vert \hat{H} \vert \phi_j \rangle} \end{gather*}\]
Geometry | $J_1$ | $J_2$ | $J_3$ | $J_4$ | $\varepsilon_1$ | $\varepsilon_2$ | $\varepsilon_{3a}$ | $\varepsilon_{3b}$ | $\varepsilon_{4}$ | RMSE |
---|---|---|---|---|---|---|---|---|---|---|
Singlet ($D_{3h}$) | -0.0382 | -0.0302 | -0.0008 | -0.0097 | 0 | 0 | 0 | 0 | 0 | $1.70\times10^{-7}$ |
Triplet ($C_{2v}$) | -0.0322 | -0.0262 | -0.0011 | -0.0081 | 0.0101 | -0.0097 | -0.0007 | 0.0003 | 0.0042 | $6.31\times10^{-9}$ |
Quintet ($C_{2v}$) | -0.0295 | -0.0240 | -0.0012 | -0.0073 | -0.0053 | -0.0095 | 0.0004 | 0.0004 | -0.0017 | $1.03\times10^{-9}$ |
Septet ($D_{3h}$) | -0.0277 | -0.0233 | -0.0012 | -0.0073 | 0 | 0 | 0 | 0 | 0 | $5.25\times10^{-7}$ |
$$ \Delta E^{(2)} \approx \sum_k \frac{(\sum_i^{\vert H_{ki}c_i \vert>\varepsilon_2} H_{ki}c_i)^2} {E^{(0)} - H_{kk}} $$
std::bitset
D.-K. Dang, J.A. Kammeraad, P.M. Zimmerman J. Phys. Chem. A 2023, 127,
std::bitset
$\implies$ int
-type
D.-K. Dang, P.M. Zimmerman J. Chem. Phys. 2021, 014105
Calculation | $\Delta E_{3-1}$ (kcal mol$^{-1}$) |
---|---|
iCASSCF (n=3) | -0.7 |
iCASSCF (n=4) | -0.6 |
iCAS-CI (n=3) | 5.1 |
iCAS-CI (n=4) | 5.5 |
$S(\zeta,n,l,m,r,\theta,\phi) = N^{\text{STO}}r^{n-1}e^{-\zeta r}Z_{lm}(\theta,\phi)$
$G(\alpha,n,l,m,r,\theta,\phi) = N^{\text{GTO}}e^{-\alpha r^2}S_{lm}(r,\theta,\phi)$
\[\begin{equation*} \hat{H} = \color{lime}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{lime}{-\frac{1}{2} \sum_i \nabla_i^2}}_{\text{e$^-$ KE}}} \color{violet}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{violet}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert} }}_{\text{e$^-$-e$^-$ repulsion}}} \color{yellow}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{yellow}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}}_{\text{nuc-e$^-$ attraction}}} \end{equation*}\]
Integrals needed:
\[\begin{gather*} \fragment{1}{(\color{yellow}{\mu \nu} \vert \color{yellow}{\lambda \kappa}) \approx \sum_{\color{violet}{PQ}} (\color{yellow}{\mu \nu} \vert \color{violet}{P}) (\color{violet}{PQ})_{\color{violet}{PQ}}^{-1} (\color{violet}{Q} \vert \color{yellow}{\lambda \kappa})} \fragment{2}{= \sum_{\color{violet}{Q}} B_{\color{yellow}{\mu\nu}}^\color{violet}{Q} B_{\color{yellow}{\lambda \kappa}}^\color{violet}{Q}} \\ \fragment{3}{B_{\color{yellow}{\mu\nu}}^\color{violet}{Q} = \sum_\color{violet}{P} (\color{yellow}{\mu \nu} \vert \color{violet}{P}) (\color{violet}{PQ})_{\color{violet}{PQ}}^{-1/2}} \\ \fragment{4}{(\color{violet}{PQ})_{\color{violet}{PQ}} = \int \int \chi_\color{violet}{P}(r_1) \frac{1}{\vert r_1 - r_2 \vert} \chi_\color{violet}{Q}(r_2) dr_1 dr_2} \end{gather*}\]
\[\begin{gather*} \fragment{1}{\int \int \chi_P(r_1) \frac{1}{\vert r_1 - r_2 \vert} \chi_\mu (r_2) \chi_\nu (r_2) dr_1 dr_2 = \int \color{yellow}{V_C^P} (r) \chi_\mu (r) \chi_\nu (r) d r} \\ \fragment{2}{\color{yellow}{V_C^P} (\zeta,n,l,m,r,\theta,\phi) = \frac{4 \pi (2 \zeta)^{n+(1/2)}}{\sqrt{(2n)!}(2l+1)}Z_{lm}(\theta,\phi)\color{lime}{I_{nl}}(r)} \\ \fragment{3}{\color{lime}{I_{nl}}(r) = r^{-l-1} \int_0^r (r^\prime)^{n+l+1} e^{-\zeta r^\prime} d r^\prime + r^l \int_r^\infty (r^\prime)^{n-l} e^{-\zeta r^\prime} d r^\prime} \end{gather*}\]
\[\begin{equation*} (\mu\nu \vert P) = N_{V_C^P,64}^{\text{STO}} N_{\chi_\mu,64}^{\text{STO}} N_{\chi_\nu,64}^{\text{STO}} \sum_i \bar{V}_C^P (x_i)_{32} \bar{\chi}_\mu(x_i)_{32} \bar{\chi}_\nu(x_i)_{32} w(x_i)_{32} \end{equation*}\]
D.-K. Dang, L.W. Wilson, P.M. Zimmerman J. Comput. Chem. A 2022 43, 1680