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

• A
• jonjacson
In summary, there are several methods used in quantum chemistry and biochemistry to calculate molecular properties and understand biochemistry, such as the Schrodinger equation, Feynman Path Integral method, and Heisenberg Matrix formulation. The most commonly used method in quantum chemistry is the Hartree-Fock method for low accuracy and coupled cluster methods for high accuracy. These methods involve solving the time-independent Schrodinger equation using matrix methods to find eigenvalues of large matrices. In biochemistry, classical molecular dynamics is typically used, using force fields fitted by the above techniques. Another important method is the Hubbard model, which can describe a variety of states in condensed matter physics. In quantum chemistry, density functional theory (DFT) is often used for non

#### jonjacson

I know to describe Quantum Mechanical systems we can use:

-Schrodinger 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!

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

bhobba and 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 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.

Last edited:
Demystifier
A. Neumaier said:
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?

jonjacson said:
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.

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.

Truecrimson and 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.

## 1. What is the difference between the Quantum Monte Carlo model and the Density Functional Theory model?

The Quantum Monte Carlo (QMC) model is a stochastic method that uses random sampling to solve quantum many-body problems. It is computationally intensive but can provide accurate results for a wide range of systems. Density Functional Theory (DFT), on the other hand, is a more efficient approach that uses a mean-field approximation to calculate the electronic density of a system. While DFT is less accurate than QMC, it is often used for larger systems due to its lower computational cost.

## 2. How does the Coupled Cluster model compare to other Quantum Models?

The Coupled Cluster (CC) model is a highly accurate and efficient method for calculating electronic properties of many-body systems. It is often referred to as the "gold standard" of quantum models due to its ability to accurately describe both weakly and strongly correlated systems. However, CC calculations can be computationally expensive and are limited to smaller systems compared to other models.

## 3. Can the Quantum Many-Body Perturbation Theory model handle strong electron-electron interactions?

Yes, the Quantum Many-Body Perturbation Theory (MBPT) model is designed to handle strong electron-electron interactions, making it suitable for studying systems with strong correlation effects. MBPT is a non-perturbative approach that uses Feynman diagrams to calculate the ground-state energy and other properties of a system. However, it can be computationally demanding and may not be suitable for large systems.

## 4. What is the advantage of using the Configuration Interaction model over other Quantum Models?

The Configuration Interaction (CI) model is a many-body approach that includes all possible configurations of electrons in a system. This makes it highly accurate for describing the electronic structure of a molecule or solid. Unlike other models, CI does not make any approximations and can accurately capture electron correlation effects. However, it is computationally expensive and is mostly limited to small systems.

## 5. Are there any quantum models that can handle both strong correlation effects and large systems?

Yes, the Density Matrix Renormalization Group (DMRG) method is a powerful approach that combines the accuracy of the Quantum Monte Carlo model with the efficiency of Density Functional Theory. DMRG is particularly useful for studying one-dimensional systems with strong electron correlation, such as quantum magnets or polymers. However, it is limited to one-dimensional systems and may not be suitable for higher dimensions.