What is the best Quantum Model to solve many body problems?

[jonjacson]

I know to describe Quantum Mechanical systems we can use:

-Schrödinger equation
-Feynman Path Integral method
-Heisenberg Matrix formulation

Well my question is, if you want to calculate molecular properties, and want to understand biochemistry (protein), you have a system with several quantum particles right?. What would be the best method to do this calculations? What is commonly used in Quantum Chemistry and Biochemistry?

Obviously I guess there are not analytical solutions to most problems so you need to use numerical methods. Is anybody familiar with Quantum Chemistry?

THanks!

 

[Demystifier]

In quantum chemistry, I think, the Schrödinger-equation approach is the most widely used method.

 

[A. Neumaier]

In quantum chemistry, one typically uses the Hartree-Fock method (for low accuracy) and coupled cluster methods (for high accuracy) to solve for the ground state energy (or in certain cases for the few lowest energy levels) of the molecular electronic Hamiltonian in dependence on the positions of the nuclei. Although the time-independent Schroedinger equation is solved, the methods are matrix methods. Everything boils down to find eigenvalus of huge matrices.

The dependence of the ground state energy on the coordinates defines the potential energy surface. This is then used to do classical molecular dynamics or quantum dynamics for the nuclei (quantum surface hopping in case several energy levels can come close). In biochemistry, one usually uses only classical molecular dynamics, using force fields fitted by the above techniques.

 

[jonjacson]

In quantum chemistry, one typically uses the Hartree-Fock method (for low accuracy) and coupled cluster methods (for high accuracy) to solve for the ground state energy (or in certain cases for the few lowest energy levels) of the molecular electronic Hamiltonian in dependence on the positions of the nuclei. Although the time-independent Schroedinger equation is solved, the methods are matrix methods. Everything boils down to find eigenvalus of huge matrices.

The dependence of the ground state energy on the coordinates defines the potential energy surface. This is then used to do classical molecular dynamics or qunatum dynamics for the nuclei (quantum surface hopping in case several energy levels can come close). In biochemistry, one usually uses only classical molecular dynamics, using force fields fitted by the above techniques.

Thanks.

By coupled cluster methods, you are not talking about the physics, it is the hardware and the numerical techniques, Am I correct?

 

[A. Neumaier]

By coupled cluster methods, you are not talking about the physics, it is the hardware and the numerical techniques, Am I correct?

No. It is an approximation technique with nontrivial physical meaning that goes beyond the Hartree-Fock approximation.
https://en.wikipedia.org/wiki/Coupled_cluster

 

[jonjacson]

Thanks.

 

[radium]

Within condensed matter physics, I would say the Hubbard model. It can describe Mott insulators, metal insulator transitions, as well as superfluid insulator transitions (the Bose Hubbard model), magnetically ordered spin states, valence bond solids, spin liquids with topological order, confinement transitions between these phases, and d wave super conductivity for example.

These states can all be described in a field theoretic framework as well. The transition in graphene from a magnetically ordered state is described by the Gross-Nevau model, the 2+1d superfluid transition by a CFT₃ (also has a duality between particles and vortices), spin liquids from slave boson/fermion constructions, etc.

In quantum chemistry, the use DFT for non-interacting system you have a density functional with spatial coordinates that determines the ground state properties you reduce a problem of N electrons to just spatial d coordinates. You can solve for the true ground state iteratively.

If the system is interacting you can add a Hubbard U. If you are studying a strongly correlated system you could use DMFT. If you want to study entangled states, you would use DMRG.

 

[jonjacson]

Within condensed matter physics, I would say the Hubbard model. It can describe Mott insulators, metal insulator transitions, as well as superfluid insulator transitions (the Bose Hubbard model), magnetically ordered spin states, valence bond solids, spin liquids with topological order, confinement transitions between these phases, and d wave super conductivity for example.

These states can all be described in a field theoretic framework as well. The transition in graphene from a magnetically ordered state is described by the Gross-Nevau model, the 2+1d superfluid transition by a CFT₃ (also has a duality between particles and vortices), spin liquids from slave boson/fermion constructions, etc.

In quantum chemistry, the use DFT for non-interacting system you have a density functional with spatial coordinates that determines the ground state properties you reduce a problem of N electrons to just spatial d coordinates. You can solve for the true ground state iteratively.

If the system is interacting you can add a Hubbard U. If you are studying a strongly correlated system you could use DMFT. If you want to study entangled states, you would use DMRG.

A lot of very useful and interesting models, thank you.