CMD III: Learning Centroid Forces
The centroid mean-force derivation, direct and delta learning in a quartic well, and two routes to molecular PIMD datasets.
Technical Notes
Lihui Gao's Technical Notes
A small place for numerical methods, quantum dynamics, programming practice, and technical study notes. I try to keep the writing reproducible, concrete, and honest about tradeoffs.
Contact me: 23110190022@m.fudan.edu.cn
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Matsubara IV: 绝热与非绝热 Matsubara 实时动力学推导
Parallel scalar and electronic-matrix derivations of the Matsubara position correlation, with cyclic traces, nuclear Gaussian integrals, and analytic consistency checks.
HEOM: Theory, Boundaries, and DVR Tests
From bath memory to the Gaussian hierarchy: its limits for original Tully SAC, a harmonic-bath DVR validation, and an explicit nuclear-basis comparison.
Reaction Computation VIII: Instantons and RPMD
Connecting stationary paths with trajectory-based rates: constrained statistical weight, signed recrossing, and a same-barrier DVR comparison.
Reaction Computation VII: Multidimensional Instantons
A coupled 2D calculation from Newton path updates to fluctuation factors and a normalized rate, with reproducible numerical checks.
Reaction Computation VI: Instanton Rates Across Crossover
Where the ordinary instanton approximation fails, what Richardson and Lawrence change, and a quantum-checked barrier benchmark across crossover.
Reaction Computation V: From Barrier Penetration to Imaginary-Time Instantons
From the dominant contribution to a rate: why instanton theory uses imaginary-time paths, tested on one barrier against independent DVR, WKB, and TST.
Reaction Computation IV: Free Energy and the Events We Rarely See
From non-Boltzmann sampling to WHAM: an analytic free-energy benchmark, and why the landscape alone does not determine a reaction time.
Reaction Computation III: When One Electronic State Is Not Enough
From 1970s surface hopping to multistate reaction dynamics: derivative couplings, a fewest-switches derivation, and a matched-ensemble quantum scattering benchmark.
Hydrogen Transfer II: From a Barrier to a Rate
A derivation-led route from Arrhenius and transition-state theory to a reproducible one-dimensional sinc-DVR rate benchmark.
Hydrogen Transfer I: What Can We Actually Observe?
Why this series begins with experiment, what hydrogen-transfer measurements can actually observe, and how microscopic calculations should meet those observables.
Learning the Generator: PINN-MMST Dynamics
A generator-learning view of MLPES and physics-informed MMST dynamics, with a reproducible small-N spin-boson benchmark and an accuracy-adjusted acceleration framework.
Codex Skills as Workflow Engineering
A controlled FSSH flowchart case study on turning successful Codex conversations into semantic contracts, deterministic tools, and evidence-based acceptance loops.
From Reactants or the Dividing Surface? An N=1 Equivalence Test
An algorithmic convergence test showing that reactant launch and dividing-surface history reconstruction estimate the same N=1 thermal flux-side rate.
Comparing Trajectory Dynamics with Kubo Flux-Side Correlations
A practical method for placing quantum Kubo and trajectory flux-side correlations on one rate-normalized scale, with a staged Tully SAC benchmark.
GPT-5.6 Reasoning for Scientific Engineering
A practical policy for choosing GPT-5.6 models, reasoning effort, subagent orchestration, and Fast processing across scientific engineering workflows.
Electronic Nearsightedness, Linear Scaling, and Local ML Potentials
A derivation-focused account of electronic nearsightedness, its consequences for linear-scaling electronic structure, and the scope and limits of locality in machine-learning interatomic potentials.
MES-PIMD and the Geometric Phase
A close reading of the MES-PI/MES-PIMD geometric-phase thermodynamics paper, with the logic chain from Jahn-Teller topology to overlap matrices, winding numbers, heat capacity, and single-state limits.
Geometric Phase: Key Theory Reproductions
Key theoretical and computational papers on molecular geometric phase effects near conical intersections, with small reproducible models for pseudorotor quantization, two-path interference, and localization.
Geometric Phase in Jahn-Teller Thermodynamics
A compact derivation of how the geometric phase changes the kinetic operator, boundary condition, degeneracy, spectrum, and heat capacity in the two-state Jahn-Teller E times e model.
How to Prepare a HefeiNAMD Training
A practical note on preparing a complete HefeiNAMD workshop with slides, server environment, precomputed outputs, Slurm scripts, pseudo jobs, and an end-to-end rehearsal path.
Jahn-Teller Instability in Li3 and Na3
A compact theoretical note on Jahn-Teller symmetry breaking, group-theoretical selection rules, conical intersections, and the triangular Li3/Na3 examples.
CMD II: Curvature Red Shift in a 2D Morse Bottle
A 2D champagne-bottle Morse benchmark showing how centroid-constrained path integrals can create an artificial-instanton basin, flatten the CMD PMF, and red-shift the radial stretch spectrum relative to direct-density-matrix DVR, RPMD, and TRPMD controls.
CMD Effective Surfaces for Tully SAC
State-resolved CMD potentials of mean force from constrained PIMC, Hermite-consistent energy-force interpolation, and SAC dynamics against DVR, Ehrenfest, FSSH, and CMD-MQC variants.
H2O Electronic Absorption with ABACUS RT-TDDFT
A fixed-nucleus H2O electronic absorption benchmark with ABACUS real-time TDDFT, aligned against the Ren, Kaxiras, and Meng molecular RT-TDDFT reference.
Perovskite Softness from Controlled NVT Spectra
Controlled 330 K NVT trajectories for MAPbI3, CsPbI3, and TiO2, analyzed through bond and angle fluctuations, displacement PCA, and MD-derived phonon density of states.
Spin Mapping and Spin-MInt Method
From MMST oscillator mapping to finite-state SU(N) spin mapping, direct Spin-MInt propagation, and Tully SAC plus three-state Morse benchmarks against FSSH, Ehrenfest, and exact DVR.
HamGNN OpenMX Workflow for TiO2 Hamiltonians
OpenMX LCAO data generation, HamGNN Hamiltonian training, band validation, and a 500 fs NVE nonadiabatic-coupling proxy check for TiO2.
Spin Mapping Mathematical Foundations
SU(N) generators, the quadratic Casimir invariant, physical versus algebraic degrees of freedom, and why Stratonovich-Weyl is the finite-state analogue of Wigner mapping.
MQC IV: MMST Mapping and ZPE Leakage
MMST oscillator mapping, electronic phase sampling, ZPE leakage, gamma choices, SQC-style corrections, and SAC diagnostics against FSSH, Ehrenfest, and DVR.
MQC III: P-Matrix Decoherence and Detailed Balance
Directed density-matrix dynamics with pairwise decoherence and Boltzmann detailed balance, tested against FSSH, Ehrenfest, and DVR toy-model diagnostics.
MQC II: Ehrenfest Mean-Field Dynamics
Mean-field electronic dynamics on Tully's simple avoided crossing, compared with a finite-grid DVR quantum reference.
MQC I: Fewest-Switches Surface Hopping
Active surfaces, hop probabilities, velocity rescaling, and a Tully simple avoided crossing benchmark with a DVR exact reference.
MQC 0: Background and Working Assumptions
A concise setup for mixed quantum-classical dynamics: BO approximation, Born-Huang couplings, NAC/HST formulas, closures, and limitations.
PIMD VI: Normal-Mode Free Propagation
Normal-mode and bead-coordinate free ring-polymer propagation compared through stability, subcycling convergence, and runtime benchmarks.
PIMD V: From PIMD to RPMD
PIMC-initialized RPMD trajectories for the harmonic-oscillator q-q Kubo correlation, with analytic limits and failure modes.
TRPMD Water IR Spectrum with q-TIP4P/F
A detailed reproduction of the q-TIP4P/F water RPMD/TRPMD infrared comparison, including PILE damping, internal-mode resonance analysis, and experimental n(alpha) overlay.
PIMD IV: Path-Integral Monte Carlo
Local Metropolis bead moves, constrained-PIMC context, and a harmonic-oscillator observable mean benchmark.
PIMD I: Ring-Polymer Basics
Path-integral isomorphism, ring-polymer equations of motion, normal modes, static estimators, and a harmonic-oscillator ensemble benchmark.
PIMD II: NVT Thermostats and GLE
PILE, GLE, PIGLET-style covariance shaping, and compact sampling diagnostics for a harmonic ring polymer.
PIMD III: Cayley Stability
Why exact normal-mode propagation can trigger resonance instabilities in ring-polymer simulations, and how the Cayley step stabilizes the free substep.
Matsubara I: From Kubo to LSC-IVR
Phase-space representation of Kubo correlations, the LSC-IVR truncation, and a harmonic/quartic benchmark.
Matsubara II: Low Modes and the Matsubara Phase
Low-frequency imaginary-time modes, the Matsubara phase, and a reproducible low-pass filtering test.
Matsubara III: Quartic Benchmark and Phase Reweighting
A quartic oscillator Kubo benchmark with HMC sampling, cosine phase reweighting, and phase-problem diagnostics.
RPMD Water IR Spectrum with SPC/F
A reproducible SPC/F fallback for the RPMD water infrared spectrum workflow, including the dipole-derivative autocorrelation estimator and drift diagnostics.
Allegro MLPES Workflow for MAPbI3
A compact MAPbI3 Allegro workflow covering data generation, extxyz packaging, training, ASE/LAMMPS deployment, and NVE validation.
DVR I: Wavepacket Dynamics Methods
Why DVR and finite-difference methods discretize the same wavepacket problem differently, with a harmonic-oscillator benchmark.
DVR II: Operator Matrices in the DVR Basis
Matrix-level intuition for kinetic, Hamiltonian, side, flux, and thermalized flux operators.
DVR III: Wavepacket Dynamics with DVR
Building a multi-state DVR Hamiltonian, diagonalizing once, and evaluating exact finite-dimensional wavepacket dynamics.
DVR IV: Density-Matrix Ensemble Evolution
Thermal initialization, positive-momentum filtering, density-matrix results, and absorber bookkeeping.
DVR V: Kubo-Transformed Position Correlation
A spectral derivation of the Kubo-transformed x-x correlation and its DVR matrix implementation.
DVR VI: Flux-Side Kubo Correlation
Side projectors, commutator-defined flux operators, CAP handling, and flux-flux integration for rate-like diagnostics.
DVR VII: Excited-Ground Kubo Population Correlation
Local adiabatic projectors, excited-channel flux selection, and ground-channel population diagnostics in a Kubo benchmark.
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Archive
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CMD III: Learning Centroid Forces
Matsubara IV: 绝热与非绝热 Matsubara 实时动力学推导
HEOM: Theory, Boundaries, and DVR Tests
Reaction Computation VIII: Instantons and RPMD
Reaction Computation VII: Multidimensional Instantons
Reaction Computation VI: Instanton Rates Across Crossover
Reaction Computation V: From Barrier Penetration to Imaginary-Time Instantons
Reaction Computation IV: Free Energy and the Events We Rarely See
Reaction Computation III: When One Electronic State Is Not Enough
Hydrogen Transfer II: From a Barrier to a Rate
Hydrogen Transfer I: What Can We Actually Observe?
Learning the Generator: PINN-MMST Dynamics
From Reactants or the Dividing Surface? An N=1 Equivalence Test
Comparing Trajectory Dynamics with Kubo Flux-Side Correlations
Electronic Nearsightedness, Linear Scaling, and Local ML Potentials
MES-PIMD and the Geometric Phase
Geometric Phase: Key Theory Reproductions
Geometric Phase in Jahn-Teller Thermodynamics
How to Prepare a HefeiNAMD Training
Jahn-Teller Instability in Li3 and Na3
CMD II: Curvature Red Shift in a 2D Morse Bottle
CMD Effective Surfaces for Tully SAC
H2O Electronic Absorption with ABACUS RT-TDDFT
Perovskite Softness from Controlled NVT Spectra
Spin Mapping and Spin-MInt Method
HamGNN OpenMX Workflow for TiO2 Hamiltonians
Spin Mapping Mathematical Foundations
MQC IV: MMST Mapping and ZPE Leakage
MQC III: P-Matrix Decoherence and Detailed Balance
MQC II: Ehrenfest Mean-Field Dynamics
MQC I: Fewest-Switches Surface Hopping
MQC 0: Background and Working Assumptions
PIMD VI: Normal-Mode Free Propagation
PIMD V: From PIMD to RPMD
TRPMD Water IR Spectrum with q-TIP4P/F
PIMD IV: Path-Integral Monte Carlo
PIMD I: Ring-Polymer Basics
PIMD II: NVT Thermostats and GLE
PIMD III: Cayley Stability
Matsubara I: From Kubo to LSC-IVR
Matsubara II: Low Modes and the Matsubara Phase
Matsubara III: Quartic Benchmark and Phase Reweighting
RPMD Water IR Spectrum with SPC/F
Allegro MLPES Workflow for MAPbI3
DVR I: Wavepacket Dynamics Methods
DVR II: Operator Matrices in the DVR Basis
DVR III: Wavepacket Dynamics with DVR
DVR IV: Density-Matrix Ensemble Evolution
DVR V: Kubo-Transformed Position Correlation
DVR VI: Flux-Side Kubo Correlation
DVR VII: Excited-Ground Kubo Population Correlation