Possible titles:

- Stochastic Methods for Molecular Properties (SMMP)
Stochastic Methods for Chiroptical Properties (SMCP or ChiroStoch)

Deterministic methods need large Hilbert spaces for effective expansions of the many-electron wave function This is however largely redundant (Ivanic 2001)

Stochastic algorithms are highly parallelizable in the number of walkers. I will develop my skills in parallel programming techniques by developing this project.

Research questions:

- Understanding chiroptical properties for large chemical systems

Objectives of the project:

The calculation of molecular properties with high accuracy and for systems of relevant size

- High accuracy means coupled cluster (CC) wave functions
- Systems for which CC is an option are limited by its polynomial scaling
- It is possible to reduce the scaling,
*e.g.*by means of local approaches to the electron correlation problem, but this has been proven to not be as effective for molecular properties as for energies

Devise the appropriate stochastic approach to the solution of response equations

- We want the stochastic approach because it's (supposedly) embarassingly parallel. This can enable the study of the response properties for larger systems and provide benchmark results for lower level calculations.
- Low scaling (or better parallelization options) + (perhaps?) controllable error of stochastic methods
are the key advantages.
**Find relevant literature on both!**citePlaceholder2*, citePlaceholder3*

The creation of the appropriate software toolbox with good scalability.

- The toolbox will be freely available under the appropriate open source license (GPL most likely!)

General background on quantum chemistry:

- "Frontiers in electronic structure theory" (Sherrill 2010)
- Quantum chemistry as an effective complement to experiment citePlaceholder5
*, citePlaceholder6*, citePlaceholder7*, citePlaceholder8*, (Goddard 1985)

State of the art:

QMC:

- Recent reviews on QMC approaches: citePlaceholder10
*, citePlaceholder11*, citePlaceholder12*, citePlaceholder13*, citePlaceholder14*, citePlaceholder15*and (Foulkes 2001) - DMC (Wick rotation of the Schrodinger equation (Wick 1954) and isomorphism with a classical diffusion problem)
- Self-Healing QMC
- Auxiliary Field QMC
- Fermion QMC
- FCIQMC
- Stochastic Coupled Cluster: Thom introduces the FCIQMC-like stochastic algorithm for solving the CC equations (Thom 2010)
The methods leverages the stochastic sampling strategies of the FCI wave function in a discrete Fock space first proposed by Alavi
*et al.*(Booth 2009) - Linked Coupled Cluster Monte Carlo: the stochastic algorithm for the solution of the CC equations in the linked (term-by-term size-extensive) form (Franklin 2016)
- Initiator approximation: the same group proposes the initiator approximation for the CCMC algorithm (Spencer 2016)
- Stochastic Møller-Plesset (Thom 2007)
- Local approaches to QMC citePlaceholder23
*, citePlaceholder24*

- Recent reviews on QMC approaches: citePlaceholder10
Properties by QMC

- Dynamic polarizabilities citePlaceholder25
*, citePlaceholder26* - Large systems citePlaceholder27
*, citePlaceholder28* - Forces: correlated sampling (Filippi 2000) and space warp coordinate transformation (Umrigar 2009) Work by Assaraf and Caffarel on improved estimators (Assaraf 2000) and (Assaraf 2003)
- Static electric properties (dipole and quadrupole moments, static polarizabilities): polarizabilities by finite differences for ethyne (Coccia 2012) polarizability of the hydrogen atom by modified sampling (Li 2007)

- Dynamic polarizabilities citePlaceholder25
Chiroptical properties:

- Eliel "Stereochemistry of organic compounds" (find appropriate ref)
- Barron's book (Barron 2004) Rosenfeld's formulation (Rosenfeld 1929)
- OR and ECD review by Pecul and Ruud (Pecul 2005)
- Berova's book (Comprehensive Chiropt...)
- Daniel's reviews citePlaceholder39
*, citePlaceholder40*, (Crawford 2012) - Octant rule (Snatzke 1979) and its failure (Rinderspacher 2004)
- Chiroptical properties by DFT citePlaceholder44
*, citePlaceholder45*, (Grimme 2001) (Stephens 2001) (Stephens 2002) (Grimme 2002) (Autschbach 2002) (Autschbach 2002a) (Autschbach 2003) - Chiroptical properties by CC (Tam 2004) (Crawford 2005) (Ruud 2002) (Ruud 2003)
- Computational studies available on a variety of molecules (Tam 2006) (Kowalczyk 2006) (Crawford 2005) (Tam 2007) (Crawford 2007) (Crawford 2007a)
citePlaceholder63
*citePlaceholder64*(Crawford 2009) (Pedersen 2009) (Pedersen 2009a) (Lambert 2012) (Rinderspacher 2004) (Wilson 2005) (Furche 2000) (Pulm 1997) (Kondru 1998) (Grimme 1998) (Polavarapu 1999) (Ribe 2000) (Polavarapu 2002) (Diedrich 2003) (Polavarapu 2003) (Norman 2004) (Diedrich 2004) (Stephens 2005) (Wiberg 2005) (Wiberg 2005a) (Wiberg 2006) (Autschbach 2009) (Pritchard 2010) (Mach 2011) (Mach 2014) - Ultrasensitive Cavity Ring-Down Polarimetry (CRDP) (Wilson 2005) (Müller 2002) (Müller 2000)
- Local approaches to the correlation problem in response theory (Russ 2004) (Russ 2008)

Response theory:

- Most recent review on wave function-based response theory (Helgaker 2012)
- Foundational work citePlaceholder96
*, citePlaceholder97*, citePlaceholder98*, citePlaceholder99* - CC response theory
- Local approaches to CC (really a lot of literature...)
- Local approaches to CC response theory citePlaceholder100
*, citePlaceholder101*, (McAlexander 2016)

Problems to address:

- The fermion sign problem. How do FCIQMC and CCMC avoid it? The sign problem is NP-hard (Troyer 2005) thus not solvable in polynomial time

TODO:

- Size of the systems investigated by stochastic methods in Fock space?
- Properties by stochastic methods? VMC/DMC/FCIQMC?
- Local correlation approaches for molecular properties?
- L. Guidoni might have published some FCIQMC calculations on large molecular systems.

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