Density Matrix Renormalization Group (DMRG)-based Approaches in Computational Chemistry

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Density Matrix Renormalization Group (DMRG)-based Approaches in Computational Chemistry
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Contents
Preface
1 Density matrix renormalization group
	1.1 Introduction
	1.2 Infinite-system density matrix renormalization group
	1.3 Finite-system density matrix renormalization group
	References
2 Tensor network states: matrix product states and relatives
	2.1 Tensor decompositions
		2.1.1 Singular value decomposition
		2.1.2 Frequently encountered tensor decompositions
	2.2 Schmidt decomposition and quantum entanglement
	2.3 Matrix product state
		2.3.1 Building matrix product state
		2.3.2 Overlaps, expectation values and matrix elements
		2.3.3 Adding two matrix product states
		2.3.4 Bringing a matrix product state into canonical form
			2.3.4.1 Generation of a left-canonical MPS
		2.3.5 Approximate compression of an MPS
		2.3.6 Good quantum numbers
	2.4 Matrix product operator
		2.4.1 Applying an MPO to an MPS
		2.4.2 Adding and multiplying MPOs
	2.5 Ground state calculations with MPS
		2.5.1 The basic algorithm
		2.5.2 Excitations
		2.5.3 “Single site” vs “two site”
		2.5.4 MPO representation of Hamiltonians
		2.5.5 Comparing DMRG to variational MPS ground state searches
	References
3 Density matrix renormalization group for semiempirical quantum chemistry
	3.1 Introduction
	3.2 Semiempirical model Hamiltonian
	3.3 Symmetrized density matrix renormalization group algorithm
		3.3.1 Particle number Ntot and Sz symmetry
		3.3.2 Spin-flip symmetry
		3.3.3 Spatial symmetry
		3.3.4 Electron-hole symmetry
	3.4 Applications
		3.4.1 The electronic structure of the ground state of cyclic polyene
		3.4.2 The excited states ordering, exciton binding, and optical properties of polyene
		3.4.3 Soliton structure of excited states of polyene
		3.4.4 Intramolecular singlet fission in donor–acceptor type conjugated copolymer
		3.4.5 Pariser–Parr–Pople density matrix renormalization group for systems beyond one-dimension
	3.5 Summary
	Acknowledgments
	References
4 Density matrix renormalization group for ab initio quantum chemistry Hamiltonian
	4.1 Renormalized operator-based density matrix renormalization group implementation
	4.2 Matrix product operator-based density matrix renormalization group implementation
	4.3 Optimal construction for matrix product operators
	4.4 Symmetries and spin adaption
	4.5 Reduced density matrix
	4.6 Orbital selection and ordering
	4.7 Error estimation
	4.8 Component analysis of density matrix renormalization group wave function
	4.9 Quantum information theory analysis
	4.10 Density matrix renormalization group for larger active spaces
	4.11 Relativistic density matrix renormalization group
	4.12 High-performance ab initio density matrix renormalization group
	4.13 Tensor network states
	References
5 Density matrix renormalization group with orbital optimization
	5.1 Orbital rotation
	5.2 The multiconfigurational self-consistent field methods
		5.2.1 Energy, gradient, and Hessian
		5.2.2 Super-configuration interaction method: a first-order multiconfigurational self-consistent field implementation
		5.2.3 Second-order multiconfigurational self-consistent field method
		5.2.4 Simultaneous optimization of configuration interaction coefficients and orbital rotations
	5.3 Density matrix renormalization group self-consistent field methods
	5.4 Excited state calculation
	5.5 Analytic gradient and geometry optimization
	5.6 Molecular spectra
	5.7 Beyond Born–Oppenheimer approximation
	5.8 Applications
		5.8.1 Electronic landscape of the P-cluster of nitrogenase
		5.8.2 Mechanism for photochromic ring-opening reaction of spiropyran
	References
6 Post-density matrix renormalization group
	6.1 Fundamentals for multireference quantum chemical calculations
		6.1.1 Static and dynamic electron correlation
		6.1.2 Contraction approximations
			6.1.2.1 Internally contracted approximations
			6.1.2.2 Externally contracted approximations
	6.2 Density matrix renormalization group-multireference configuration interaction
		6.2.1 Density matrix renormalization group-fully internally contracted-multireference configuration interaction
		6.2.2 Density matrix renormalization group-externally contracted-multireference configuration interaction
		6.2.3 Uncontracted matrix product state-multireference configuration interaction
	6.3 Density matrix renormalization group-multireference perturbation theory
		6.3.1 Recapitulation of multireference perturbation theory
			6.3.1.1 Rayleigh–Schrödinger perturbation theory
			6.3.1.2 Different perturbation theory partitions
				6.3.1.2.1 Complete active space with second-order perturbation theory
				6.3.1.2.2 ENPT2
				6.3.1.2.3 Second-order N-electron valence state perturbation theory
				6.3.1.2.4 Linearized coupled cluster doubles
		6.3.2 Density matrix renormalization group-complete active space with second-order perturbation theory
		6.3.3 Density matrix renormalization group-second-order N-electron valence state perturbation theory
		6.3.4 Density matrix renormalization group-ENPT2
		6.3.5 Matrix product states-perturbation theory
			6.3.5.1 Matrix product states-linearized coupled cluster
		6.3.6 Other variants
	6.4 Density matrix renormalization group-coupled cluster theory
		6.4.1 Recapitulation of coupled cluster theory
		6.4.2 Density matrix renormalization group-alternative-multireference coupled cluster
		6.4.3 Density matrix renormalization group-canonical transformation
	6.5 Hybridization of density matrix renormalization group with density functional theory
		6.5.1 Recapitulation of density functional theory
		6.5.2 Density matrix renormalization group-short-range density functional theory
		6.5.3 Density matrix renormalization group-pair density functional theory
	6.6 Density matrix renormalization group-adiabatic connection
	6.7 Embedding density matrix renormalization group in environments
		6.7.1 Density matrix renormalization group-in-density functional theory
		6.7.2 Polarizable embedding density matrix renormalization group
		6.7.3 Combining density matrix renormalization group with reference interaction site model
	6.8 Summary and outlook
	References
7 DMRG in frequency space
	7.1 Introduction
	7.2 Spectral function in linear response regime
	7.3 Algorithms at zero temperature
		7.3.1 Lanczos density matrix renormalization group
			7.3.1.1 The multi-targeting scheme and adaptive scheme
			7.3.1.2 The matrix product states/matrix product operators scheme
		7.3.2 Correction vector density matrix renormalization group
		7.3.3 Dynamical density matrix renormalization group
		7.3.4 Chebyshev matrix product states
		7.3.5 Analytic linear response density matrix renormalization group
	7.4 Finite temperature algorithms
		7.4.1 Lanczos density matrix renormalization group
		7.4.2 Dynamical density matrix renormalization group
		7.4.3 Chebyshev matrix product state
	7.5 Applications
		7.5.1 Electron system
		7.5.2 Electron–phonon system
	7.6 Summary and outlook
	References
	Further reading
8 Time-dependent density matrix renormalization group
	8.1 Overview
		8.1.1 Time-dependent density matrix renormalization group and nonadiabatic dynamics
		8.1.2 Relation between time-dependent density matrix renormalization group and multilayer multiconfiguration time-dependent...
		8.1.3 Reviews, software, and other resources
	8.2 Time evolution algorithms
		8.2.1 Propagation and compression
		8.2.2 Time-dependent variational principle
		8.2.3 Time step targeting
	8.3 Finite temperature algorithms
		8.3.1 Purification in an enlarged Hilbert space
		8.3.2 Minimally entangled typical thermal states
	8.4 Applications
		8.4.1 Exciton and charge transfer dynamics
		8.4.2 Excited state dynamics and spectra
		8.4.3 Charge transport
		8.4.4 Electron dynamics
	8.5 Summary and outlook
	References
Index
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