# Far down the Quantum Chemistry: a Proposal to Introduce an “Entangled” Electron Correlation

## Exceeding the Old Methods

In __ Quantum Chemistry__, the

__(__

**Hartree–Fock**__)__

**HF**

**Method**^{a}, which describes

**interacting**

**Fermion****systems**using an

**effective Single-Particle Model**[

__1__], is

**widely**used to

**approximate**the

__of__

**Electronic Structure****atoms**and

**molecules**.

However, this method **neglects** __ Electron Correlation__, which results from the inherent

**interactions**between

__s in quantum systems. The__

**Electron****Correlation Energy**[

__2__], defined as the

**energy difference**between the

**HF limit**and the

**exact**

**solution**of the

**) (further information in**

**nonrelativistic Schrödinger Equation**(__Equation 1____, Section 2), is one measure of this correlation.__

**here****Equation 1**. The Electronic Energy in HF Method

Nevertheless, other **metric**s, such as **Statistical Correlation Coefficient**s [__3__] and __ Shannon Entropy__, have been proposed to

**quantify electron correlation**. Despite these methods, electron correlation remains challenging to calculate

**accurately**for

**Large System**s [

__4__].

**Quantum Entanglement** (further information in ** here**, Ref. 2), a fundamental concept in

**quantum mechanics**, offers a

**promising alternative**for

**measuring**electron correlation. Unlike

**traditional measures**, entanglement is

**directly observable**and represents a

**non-classical correlation**between quantum systems.

## A New Definition of Quantum Entanglement

To **quantify** entanglement, one can consider a **pure two-electron state** in a **2m-dimensional** **Spin-Orbital**^{b} (__Figure 1__) space, represented by **fermionic** __ Annihilation__ and

**Creation****Operator**s (further information in

**, Section 2), in the order, [math]\normalsize{c_{a(b)}}[/math] and [math]\normalsize{{c^{\dagger}}_{a(b)}}[/math], with [math]\small{|0 \rangle}[/math] as the**

__here__**vacuum state**. The

**general form**of a two-electron state, [math]\small{| \Psi \rangle}[/math] can be written as:

**Equation 2**. The form of the Electronic Wavefunction for a two-electron state

where [math]\normalsize{x_{a, b}}[/math] represents the **Antisymmetric****Expansion Coefficient Matrix** [__5__], satisfying [math]\normalsize{x_{a, b} = - x_{b, a}}[/math]. Using **this representation**, we can derive a **Reduced Density Matrix** (further information in ** here**, Section 2), [math]\normalsize{\rho}[/math] by tracing out all but

**one Spatial Orbital**

^{b}, resulting in a [math]\normalsize{4 \times 4}[/math] matrix. Then one can define the

__, like below.__

**Von Neumann Entropy****Equation 3**. The Von Neumann Entropy equation

With [math]\normalsize{\rho}[/math] of the following form:

**Equation 4**. The expansion of the Reduced Density Matrix

This approach provides a **measure of the entanglement** for **atomic** and **molecular systems**, **focusing** on the von Neumann entropy of the reduced density matrix.

**Figure 1**. A 3D Sphere constitutes a typical Illustration of a 1s Atomic Orbital

## Is it an Efficient Model?

Using the derived reduced density matrix, the entanglement for various systems, focusing on the **Hydrogen Molecule** ([math]\small{H_{2}}[/math]) [__6__], as an example, can be calculated. The entanglement is evaluated as a **function** of the __ Interatomic Distance__,

__.__

*R*These calculations show that entanglement (__Equation 6__) and electron correlation exhibit **similar trends**, with **maximum** entanglement occurring at **specific interatomic distances**. This behavior aligns with previous findings, indicating that entanglement can be an **effective metric** for measuring electron correlation.

Additionally, if one explores a model system of two spin-1/2 electrons with an **Exchange Coupling Constant** ^{a}, [math]\small{J}[/math] and a **transverse** **Magnetic Field****strength**, __B__. The general

__for this system is given by:__

**Hamiltonian****Equation 5**. The Hamiltonian form for a two-electron state

where the subscripts, [math]\small{1}[/math] and [math]\small{2}[/math], stand for the **two electrons**, respectively; [math]\normalsize{\sigma^{x}}[/math], [math]\normalsize{\sigma^{y}}[/math] and [math]\normalsize{\sigma^{z}}[/math] are the __ Pauli Matrices__ in the three

**Cartesian directions**([math]\normalsize{x}[/math], [math]\normalsize{y}[/math] and [math]\normalsize{z}[/math]) while [math]\normalsize{\gamma}[/math] is the

**Degree**of

__([math]\small{I}[/math]). This model provides a simplified framework for examining entanglement in a quantum system.__

**Anisotropy****Equation 6**. The Quantum Entanglement form expressed via Von Neumann Entropy equation

[math]\large{\lambda}[/math] represents the __ Eigenvalue__ for the Hamiltonian of the

**two spin system**(further information in

**, Section 3).**

__here__## A Cross-cutting Solution with no Traces of Complexity...

The results demonstrate that **quantum entanglement** can serve as an **effective measure** of **electron correlation** in **quantum chemistry**. The use of **von Neumann entropy** allows for a **more observable** and **intuitive understanding** of electron correlation, without relying on **traditional methods** that require **complex wave function calculations**.

**This approach** has **implications** for **larger atomic** and **molecular systems** and can be extended to other quantum systems, offering a **robust alrternative** for evaluating **electronic structures** in atoms and molecules. Future work will explore the application of this method to **more complex systems** and its **potential** for **advancing quantum chemistry calculations**.

- ResearchForLife7 (revisited from Springer). "What Is a Single- Particle Model?"
__https://httpsresearchforlife7.com/wp-content/uploads/2024/05/What_Is_a_Single___Particle_Model_.pdf__ - ACS Pubblications. "Dynamic and Nondynamic Electron Correlation EnergyDecomposition Based on the Node of the Hartree−Fock SlaterDeterminant"
__https://pubs.acs.org/doi/epdf/10.1021/acs.jctc.3c00828__ - Core. "Statistical angular correlation coefficients
and second electron-pair moments for atoms"
__https://core.ac.uk/download/pdf/59119911.pdf__ - Nature. "Towards accurate quantum simulations of large systems with small computers"
__https://www.nature.com/articles/srep41263__ - American Physical Society. "Structure of Fermionic Density Matrices: Complete NRepresentability Condition"
__https://link.aps.org/accepted/10.1103/PhysRevLett.108.263002__ - Knowino. "Hydrogen - like atom"
__https://www.theochem.ru.nl/~pwormer/Knowino/knowino.org/wiki/Hydrogen-like_atom.html__