A Primer on Quantum Mechanics

• First postulate: A system is completely specified by its wave function, $\psi$
• Second postulate: A classical observable, $O$, may be obtained by applying Hermitian operator, $\hat{O}$, to the wave function
• $$\hat{O}\vert \psi \rangle = O \vert \psi \rangle$$

The Schrödinger Equation

\[\begin{equation*} \hat{H} \vert \psi \rangle = E \vert \psi \rangle \end{equation*}\]

The Hamiltonian

\[\begin{gather*} \fragment{1}{\hat{H}} \fragment{2}{= \color{lime}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{lime}{-\frac{1}{2} \sum_i \nabla_i^2}}_{\text{e$^-$ KE}}}} \fragment{3}{ \color{cyan}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{cyan}{-\frac{1}{2} \sum_I \nabla_I^2}}_{\text{nuc KE}}}} \fragment{4}{ \color{violet}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{violet}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert} }}_{\text{e$^-$-e$^-$ repulsion}}}} \fragment{5}{ \color{orange}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{orange}{+ \sum_{I>J} \frac{Z_I Z_J}{\vert R_I - R_J \vert} }}_{\text{nuc-nuc repulsion}}}} \fragment{6}{ \color{yellow}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{yellow}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}}_{\text{nuc-e$^-$ attraction}}}} \end{gather*}\]

The Hamiltonian

\[\begin{gather*} \hat{H} = \color{lime}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{lime}{-\frac{1}{2} \sum_i \nabla_i^2}}_{\text{e$^-$ KE}}} \color{cyan}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\boxed{\color{cyan}{-\frac{1}{2} \sum_I \nabla_I^2}}}_{\text{nuc KE}}} \color{violet}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{violet}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert} }}_{\text{e$^-$-e$^-$ repulsion}}} \color{orange}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\boxed{\color{orange}{+ \sum_{I>J} \frac{Z_I Z_J}{\vert R_I - R_J \vert} }}}_{\text{nuc-nuc repulsion}}} \color{yellow}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{yellow}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}}_{\text{nuc-e$^-$ attraction}}} \end{gather*}\]

The Electronic Hamiltonian

\[\begin{equation*} \hat{H} = \color{lime}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{lime}{-\frac{1}{2} \sum_i \nabla_i^2}}_{\text{e$^-$ KE}}} \color{violet}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{violet}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert} }}_{\text{e$^-$-e$^-$ repulsion}}} \color{yellow}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{yellow}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}}_{\text{nuc-e$^-$ attraction}}} \end{equation*}\]

$$\hat{H} \vert \psi \rangle = E \vert \psi \rangle$$ Only solvable for H-atom

The Variational Principle

\[\begin{equation*} \langle \psi \vert \hat{H} \vert \psi \rangle \geq E_0 \end{equation*}\]

Guess Wave function $\implies$ Optimize

Basis Selection

• Project $\psi$ onto basis
• Atom-centered, 1-electron basis
• Hydrogen-like atomic wave functions : atomic orbitals
• Take linear combination of atomic orbitals (LCAO) to form molecular orbitals (MO)

LCAO-MO

Functional form of the Wave Function

Naive approach \[\begin{gather*} \fragment{2}{\psi=\phi_1(\mathbf{r}_1)\phi_2(\mathbf{r}_2)\cdots\phi_N(\mathbf{r}_N)} \fragment{3}{\impliedby \text{No exchange}} \\\\ \fragment{4}{\psi = \frac{1}{\sqrt{N!}} \begin{vmatrix} \phi_1(\mathbf{r}_1) & \phi_2 (\mathbf{r}_1) & \cdots & \phi_N(\mathbf{r}_1) \\ \phi_1(\mathbf{r}_2) & \phi_2 (\mathbf{r}_2) & \cdots & \phi_N(\mathbf{r}_2) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_1(\mathbf{r}_N) & \phi_2 (\mathbf{r}_N) & \cdots & \phi_N(\mathbf{r}_N) \end{vmatrix} } \fragment{5}{\impliedby \text{has exchange}}\\\\ \fragment{6}{\implies\text{Optimize orbitals}} \end{gather*}\]

Electron Correlation

• HF (single determinant) only qualitatively correct (sometimes)
• Missing electron correlation

Configuration Interaction

\[\begin{gather*} \fragment{4}{\hat{H}\psi = E \psi} \fragment{5}{\implies \mathbf{H}\mathbf{C} = E\mathbf{C}} \\ \fragment{6}{ H_{ij} = \langle \phi_i \vert \hat{H} \vert \phi_j \rangle} \end{gather*}\]

Full CI Scaling

• 1.6 EFLOPS $\implies 1.6 \times 10^{18}$ FLOPS
• At C$_4$H$_6$ need >$3$ yrs for single row of $\mathbf{HC}$ matrix-vector multiplication

Non-exact correlated methods

• CISD (only include single and double excitations) scales $O(N^6)$
• CCSD scales $O(N^6)$
• CCSD(T) scales $O(N^7)$: considered "Gold Standard"

Complete Active Space

\[\begin{gather*} {\hat{H}\psi = E \psi} {\implies \mathbf{H}\mathbf{C} = E\mathbf{C}} \\ { H_{ij} = \langle \phi_i \vert \hat{H} \vert \phi_j \rangle} \end{gather*}\]

Restricted Active Space

An Open-Shell Coronoid

An Open-Shell Coronoid

An Open-Shell Coronoid

Coronoid Computational Methods

• Optimize structures with CASSCF(6e,6o)/cc-pVDZ
• Compute spin states with RAS(6,6)-SF/cc-pVDZ

Coronoid Energies

Polyradicaloid indices

Odd electron density

CI revisited

• CI Hamiltonian elements are mostly zero
• $$ \fragment{1}{\hat{H}} \fragment{2}{= \color{cyan}{-\frac{1}{2} \sum_i \nabla_i^2} \color{cyan}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}} \fragment{3}{\color{yellow}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert}}} $$

Heat-bath CI

• First: get a variational wave function
• Select important determinants

Heat-bath CI

• Second step: do perturbation theory
• Recover remaining electron correlation

$$ \Delta E^{(2)} \approx \sum_k \frac{(\sum_i^{\vert H_{ki}c_i \vert>\varepsilon_2} H_{ki}c_i)^2} {E^{(0)} - H_{kk}} $$

HCI implementation details

• Encode Slater determinants: 1-hot encode $\alpha$ and $\beta$ electrons
• Store $\alpha$ and $\beta$ strings in std::bitset

D.-K. Dang, J.A. Kammeraad, P.M. Zimmerman J. Phys. Chem. A 2023, 127, 1, 400

Parallelization of HCI

• Hash $\alpha$ and $\beta$ strings: std::bitset $\implies$ int-type

#define FNV_32_PRIME 0x0100093
#define MAXORBS 192
__attribute__((optimize("unroll-loops")))
size_t Fnv_hash_bytes(const void* ptr)
{
	size_t hash = FNV_32_PRIME;
	const size_t* cptr = static_cast<const size_t*>(ptr);
	int len = MAXORBS / 64;
	for (;len;--len)
	{
		hash ^= static_cast<size_t>(*cptr++);
		hash *= static_cast<size_t>(2870177450012600261ULL);
	}
	hash = hash >> 37;
	return hash;
}
#define FNV_32_PRIME 0x0100093
#define MAXORBS 192
__attribute__((optimize("unroll-loops")))
size_t Fnv_hash_bytes(const void* ptr)
{
	size_t hash = FNV_32_PRIME;
	const size_t* cptr = static_cast<const size_t*>(ptr);
	int len = MAXORBS / 64;
	for (;len;--len)
	{
		hash ^= static_cast<size_t>(*cptr++);
		hash *= static_cast<size_t>(2870177450012600261ULL);
	}
	hash = hash >> 37;
	return hash;
}

HCI Performance

• Butadiene (C$_4$H$_6$) in cc-pVDZ
• (22e,82o) - ~$10^{26}$ FCI determinants
• $3.2 \times 10^6$ variational determinants
• $2.7 \times 10^{10}$ perturbative determinants

Cyclobutadiene Automerization with HCI

• C$_4$H$_4$ with cc-pVTZ
• (20e,172o) - ~$2\times10^{31}$ FCI determinants

[$\text{FeO}(\text{NH}_3)_5$]$^{2+}$

• Fe, O: cc-pVTZ, N: cc-pVDZ, H: 6-31G
• (82e,198o) all electron active space - $3 \times 10^{85}$ FCI determinants
• Truncate to (22e,168o) - $3\times 10^{33}$ FCI determinants

[$\text{FeO}(\text{NH}_3)_5$]$^{2+}$

[$\text{FeO}(\text{NH}_3)_5$]$^{2+}$

Strategies For Big CI

• Many-body expansion $$E = E_{ref} + \sum_{i} \varepsilon_i + \sum_{i>j} \varepsilon_{ij} + \sum_{i>j>k} \varepsilon_{ijk} + \cdots$$
• Chemically intuitive definition of "bodies"

The many-body expansion for CAS

D.-K. Dang, P.M. Zimmerman J. Chem. Phys. 2021, 014105

$\text{i}$CASSCF

\[\begin{gather*} \nabla E^o \rightarrow 0 \\ ^o E_{pq}^{(1)} = 2 (F_{pq} - F_{qp}) \rightarrow 0 \end{gather*}\]

• $F_{pq}$ looks like single excitation from $p \rightarrow q$
• Numerical trick: remove redundant single excitations

$\text{i}$CASSCF Energy Benchmarks

$\text{i}$CASSCF Geometry Benchmarks

$\text{oxoMn(salen)Cl}$

• (84e,84o)
• $2.5 \times 10^{48}$ FCI determinants

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

Basis Selection Revisted

Gaussian Orbitals

$S(\zeta,n,l,m,r,\theta,\phi) = N^{\text{STO}}r^{n-1}e^{-\zeta r}Z_{lm}(\theta,\phi)$

$G(\alpha,n,l,m,r,\theta,\phi) = N^{\text{GTO}}e^{-\alpha r^2}S_{lm}(r,\theta,\phi)$

Molecular Integrals

\[\begin{equation*} \hat{H} = \color{lime}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{lime}{-\frac{1}{2} \sum_i \nabla_i^2}}_{\text{e$^-$ KE}}} \color{violet}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{violet}{+ \sum_{i>j} \frac{1}{\vert r_i - r_j \vert} }}_{\text{e$^-$-e$^-$ repulsion}}} \color{yellow}{\underbrace{\vphantom{\boxed{\sum_{i>j}}}\color{yellow}{- \sum_I \sum_i \frac{Z_I}{\vert r_i - R_I \vert}}}_{\text{nuc-e$^-$ attraction}}} \end{equation*}\]

Integrals needed:

• $\color{lime}{\int \chi_p (r) \frac{1}{2} \nabla^2 \chi_q (r) dr = (p\vert \frac{1}{2} \nabla^2 \vert q)}$
• $\color{violet}{\int \int \chi_p (r_1) \chi_q (r_1) \frac{1}{\vert r_1 - r_2 \vert} \chi_r (r_2) \chi_s (r_2) dr_1 dr_2 = (p q \vert r s)}$
• $\color{yellow}{\int \chi_p (r) \frac{Z_I}{\vert R_I - r \vert} \chi_q (r) dr = (p \vert \frac{Z_I}{\vert R_I - r \vert} \vert q)}$
• $\int \chi_p(r) \chi_q(r) dr = (p \vert q)$

Resolution of the Identity Approximation

• Condenses 4-center integrals to 2- and 3-center integrals
• Utilizes an auxiliary basis to fit the density

\[\begin{gather*} \fragment{1}{(\color{yellow}{\mu \nu} \vert \color{yellow}{\lambda \kappa}) \approx \sum_{\color{violet}{PQ}} (\color{yellow}{\mu \nu} \vert \color{violet}{P}) (\color{violet}{PQ})_{\color{violet}{PQ}}^{-1} (\color{violet}{Q} \vert \color{yellow}{\lambda \kappa})} \fragment{2}{= \sum_{\color{violet}{Q}} B_{\color{yellow}{\mu\nu}}^\color{violet}{Q} B_{\color{yellow}{\lambda \kappa}}^\color{violet}{Q}} \\ \fragment{3}{B_{\color{yellow}{\mu\nu}}^\color{violet}{Q} = \sum_\color{violet}{P} (\color{yellow}{\mu \nu} \vert \color{violet}{P}) (\color{violet}{PQ})_{\color{violet}{PQ}}^{-1/2}} \\ \fragment{4}{(\color{violet}{PQ})_{\color{violet}{PQ}} = \int \int \chi_\color{violet}{P}(r_1) \frac{1}{\vert r_1 - r_2 \vert} \chi_\color{violet}{Q}(r_2) dr_1 dr_2} \end{gather*}\]

6D integration to 3D integration

• Use a central potential for a Slater orbital $$ S(\zeta,n,l,m,r,\theta,\phi) = N^{\text{STO}}r^{n-1}e^{-\zeta r}Z_{lm}(\theta,\phi) $$

\[\begin{gather*} \fragment{1}{\int \int \chi_P(r_1) \frac{1}{\vert r_1 - r_2 \vert} \chi_\mu (r_2) \chi_\nu (r_2) dr_1 dr_2 = \int \color{yellow}{V_C^P} (r) \chi_\mu (r) \chi_\nu (r) d r} \\ \fragment{2}{\color{yellow}{V_C^P} (\zeta,n,l,m,r,\theta,\phi) = \frac{4 \pi (2 \zeta)^{n+(1/2)}}{\sqrt{(2n)!}(2l+1)}Z_{lm}(\theta,\phi)\color{lime}{I_{nl}}(r)} \\ \fragment{3}{\color{lime}{I_{nl}}(r) = r^{-l-1} \int_0^r (r^\prime)^{n+l+1} e^{-\zeta r^\prime} d r^\prime + r^l \int_r^\infty (r^\prime)^{n-l} e^{-\zeta r^\prime} d r^\prime} \end{gather*}\]

Integrating Slater Orbitals on a grid

• Lebedev angular grid
• Log radial grid
• Voronoi polyhedra partitioning and weighting

\[\begin{equation*} (\mu\nu \vert P) = N_{V_C^P,64}^{\text{STO}} N_{\chi_\mu,64}^{\text{STO}} N_{\chi_\nu,64}^{\text{STO}} \sum_i \bar{V}_C^P (x_i)_{32} \bar{\chi}_\mu(x_i)_{32} \bar{\chi}_\nu(x_i)_{32} w(x_i)_{32} \end{equation*}\]

D.-K. Dang, L.W. Wilson, P.M. Zimmerman J. Comput. Chem. A 2022 43, 1680

GPU Performance

Multi-GPU efficiency

Cyclobutadiene automerization in STO basis

STO Gradients

Summary

• With CASSCF and RAS-SF, handle problems where electron correlation is localized to a small number of electrons (12 for routine use)
• Using HCI can reduce the computational burden, and extrapolate to FCI
• With computational advances and parallelization, can treat 22 electrons with FCI-level accuracy
• Using the many-body expansion, get to 84 electrons
• With RI and GPUs, can utilize the more accurate STO basis

Acknowledgments

Committee members

• Paul Zimmerman
• Eitan Geva
• Robert Krasny
• Roseanne Sension

Zimmerman Group

• Electronic Structure
  Subgroup
• Leighton Wilson
• Josh Kammeraad

Gavini Group

• Bikash Kanungo
• Ian Lin
• Sambit Das

External Collaborators

• Marcin Stepien
• Slava Bryantsev

Cluster support

• David Braun

Funding

• NSF GRFP
• MICDE
• DOE
• UM Chemistry
• UM Rackham

Computational
Resources

• XSEDE, SDSC
  (Expanse)
• DOE, NERSC
  (Perlmutter)