Knowdes
A dual vision on geminals
Introduction
The hole vacuum - hole operators
Particle-hole duality in geminal wave function forms
Algebraic geminal creators
Algebraic structures
Algebras
Analytical Jacobi rotations for AP1roG
Angular momentum
AP1roG PSEs
APIG PSEs
Arc length
Argument for spin-separated spinor bases
Bi-orthogonal operators
Bibliography
Bilinear maps
Biot-Savart’s law
Calculating nuclear shieldings
Common integrals
Orbital optimization through conjugate pairs
Complex orbital optimization through real parameters
Conceptual DFT
Condensed Fukui function
Mulliken’s fragment of molecular response approach (FMR)
Configuration interaction
Configuration state functions
Contractions between Thouless transformed operators
Contractions of AO basis operators in an asymmetric Fermi vacuum
Single contractions
Multiple contractions
Constraining wave function models
Coulomb gauge
Covariant electrodynamics
Creation and annihilation operators
Curl
Density matrices in a restricted formalism
Density matrices
Derivation of the GHF SCF equations through Lagrange multipliers
Derivation of the UHF SCF equations through Lagrange multipliers
Derivatives over London orbitals
DIIS
Dipole moments
Dirac-Hartree-Fock theory
The Hartree-Fock wave function
The Dirac-Hartree-Fock equations
Dirac-Hartree-Fock (Coulomb) theory in terms of 2-spinors
Dirac-Hartree-Fock theory in terms of 4-spinors
Directional derivatives
Divergence
Divergence theorem
DOCI
Doubly-occupied configurations
The doubly-occupied Hamiltonian
DOCI Hamiltonian matrix elements
DOCI matrix-vector products
Density matrices for DOCI
Edmiston-Ruedenberg localization
Electron pair operators
Elementary anticommutation relations
Entropy measures for wave functions
Examples of groups
Expectation values over Fockian operators
In a general spinor basis
In a restricted spin-orbital basis
Extended Thouless theorem
FCI
Fermionic field operators
General spinor basis
Spin-orbital basis
Fields
First-order projected Schrödinger equation response properties
First-order response derivatives for the variational Lagrangian
First-order response derivatives for variationally determined wave functions
Fock space
Fockian operators
In a general spinor basis
In a ‘restricted’ spin-orbital basis
Formulas for commutators and anticommutators
Foster-Boys localization
Fukui function
Frozen-core approximation
Interpretation of the Fukui function
Fukui matrix
Physical interpretation
Functionals
Functional derivative
Calculational properties
Functions
Gauge transformations
Gaussian-type orbitals
Normalization
Geminal pairing schemes
Geminal wave functions
APG, APSG and APIG
Some more commutators
AGP
Simplifications
The AP1roG wave function
The APr2G wave function
General Hartree-Fock theory
General spinor bases
General spinor rotations
Generalized Slater-Condon rules
One-body operators
Two-body operators
Gradient
Group homomorphisms and isomorphisms
Groups
Hermite Gaussians
Induced molecular current densities
Integrals over London orbitals
Jacobi rotations
Kinematic effects
Lie algebras
Limits
Line integral
Scalar functions
Vector functions
Linear maps
Linear response derivatives for CI
Linear response derivatives for GHF
External magnetic field without London orbitals
Linear response derivatives for RHF
Electric response derivatives
External magnetic fields without London orbitals
Gauge transformations (without London orbitals)
London orbitals
Lorentz transformations
Lorenz gauge
Magnetic molecular properties
Matrix exponentials
Maxwell’s equations
The standard Maxwell equations
Maxwell’s equations for the potentials (Lorenz gauge)
Maxwell’s equations for the potentials (Coulomb gauge)
Minimizing weighted overlap orthonormalization
Formulation of the problem
Solving the problem
A generalization
Minkowski space
Molecular response properties
Möller-Plesset perturbation theory
Morphisms
Motivation for wave functions built from electron pairs
Mulliken partitioning
Mulliken population analysis
N-DMs
Open research questions
\(N\)
-DMs for spin orbitals
Theory
Implementation
N-electron probability densities and current densities
Non-orthogonal CI
Non-orthogonal spinor bases
Nuclear shielding
Occupation number operators
One-electron excitation operators
One-electron probability densities and probability current densities
Orbital index conventions
Orbital occupation number operators
Orbital optimization in the projected Schrödinger equation framework
Orbital optimization
Overlap between ONVs in different general spinor bases
Overlap between ONVs in restricted and unrestricted spin-orbital bases
Perturbation-dependent Fock spaces
The perturbation affects the metric
The perturbation does not affect the scalar metric}
Probability and charge densities
Probability density and probability current density operators
Quantizing one- and two-electron operators in general spinor bases
Quantizing one-electron operators in general spinor bases
Quantizing operators in spin-orbital bases
‘Unrestricted’ spin-orbital bases
Restricted spin-orbital bases
Quantizing relativistic one- and two-electron operators
Relativistic fermionic field operators
Quantizing the Dirac Hamiltonian
Quantizing the two-electron interactions
Towards a second-quantized four-component quasi-relativistic many-particle Hamiltonian
Avoiding variational collapse
Current densities for many electrons
Self-consistent field equations
Quantizing two-electron operators in general spinor bases
Quantum mechanical states
Real singlet orbital optimization in the projected Schrödinger equation framework
Reduced density matrices
Reduced Hamiltonians
Redundant orbital rotation generators in GHF
Relating nuclear shieldings with induced current densities
Relativistic Lagrangians for interactions with electromagnetic fields
The relativistic Lagrangian for a free particle
The interaction between a
\(4\)
-current and the electromagnetic field
A charged particle subject to an electromagnetic field
The interaction of two moving charged particles
Representation theory
Response density matrices
Restricted excitation operators
Restricted Hartree-Fock theory
Restricted orbital optimization
Restricted spin operators
Restricted spin-orbital bases
Ritz pairs
Rotations of states, operators and expectation values
Second-order optimization for GHF
Through conjugate pairs
Through real parameters
Projected Schrödinger equation second order response properties
Second-order properties
Second-order response derivatives for the variational Lagrangian
Second-order response derivatives for variationally determined wave functions
Second-quantized density and current density operators
In general spinor bases
In a restricted spin-orbital basis
Second-quantized Hamiltonians
Second-quantized operator products
Seniority-preserving operators
An alternative derivation of the seniority-preserving part of the Hamiltonian
Spin-separated parts of seniority-preserving operators
Raising and lowering operators
Seniority raising and lowering operators
Open questions and ideas
Seniority
Sets
Solving systems of equations
Spin contamination in UHF
Spin expectation values for GHF
Spin expectation values for RHF
Spin expectation values for UHF
Spin in general spinor bases
Spin in spin-orbital bases
Unrestricted spin-orbital bases
Restricted spin-orbital bases
Spin operators
Spin-orbital bases
Spin phase transitions
Spin tensor operators
Spinor transformations
Splitting up sums
Surface integrals
Surface integrals of scalar fields
Surface integrals of vector fields
The 1-DM
The 2-DM
The Broyden method for systems of equations
The correspondence principle
The coupled-cluster wave function model
The Davidson diagonalization method
The Davidson-Liu diagonalization method
The effective spin-orbit operator
The electrical dipole moment and polarizability
The electron pair density operator
The generalized Gaudin algebra
The GHF magnetic inducibility
The Hermitian adjoint of a Pauli operator
The Hessian
The inactive Fock matrix
The Lie algebra su(2)
The linear response equations for the variational Lagrangian
The linear response equations for variationally determined wave functions
The linear variation method
The magnetic inducibility as a response property
The McMurchie-Davidson integral scheme
Overlap integrals
Linear momentum integrals
Kinetic energy integrals
Dipole/position integrals
Angular momentum integrals
The molecular magnetic Hamiltonian
The Newton method for scalar optimizations
The Newton method for systems of equations
The Olsen addressing scheme
The one-electron Pauli Hamiltonian
The one-electron Schrödinger Hamiltonian
The one-electron Schrödinger Hamiltonian for a free electron
The one-electron Schrödinger Hamiltonian for an electron in an electromagnetic field
The orbital rotation generator
The Pauli matrices
The Pauli overlap operator
The projected Schrödinger coupled-cluster equations
The projected Schrödinger equation framework
The projected Schrödinger Lagrangian
The Rayleigh energy function
The relativistic electron
The Klein-Gordon equation
The Dirac equation
The standard representation
Other representations
Negative-energy states
The Dirac electron in an external electromagnetic field
The non-relativistic limit - the Pauli equation
The time-independent Dirac equation
The RHF magnetic inducibility
Without London orbitals
Derivation
Continuous set of gauge transformations
Connection with
(Keith and Bader 1993)
The Schrödinger equation
The second-quantized Pauli Hamiltonian
The Slater-Condon rules
The spin-orbital rotation generator
Unrestricted spinor rotations
Restricted spinor rotations
The symmetric eigenvalue problem
The variational Lagrangian
Thouless theorem
Thouless transformed operators
Topology
Total derivatives
Two-electron excitation operators
Two-electron integrals
Uniform electromagnetic fields
Uniform electric fields
Uniform magnetic fields
Unit normal vector
Units
Unrestricted Hartree-Fock
Variationally-determined wave functions
Vector calculus
Vector fields
Vector spaces
Wave function models
Common integrals