# Far down the Quantum Chemistry: how 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**(

__Equation 1__) of the

**nonrelativistic Schrödinger Equation**(further information in

**, Section 2), is one**

__here__**measure**of this

**correlation**.

**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__ space, represented by

**fermionic**

__and__

**Annihilation**

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

**, Section 2), [math]\normalsize{c_{a}}[/math] and [math]\normalsize{{c^{\dagger}}_{a}}[/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**, 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.

## 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], are the **two electrons**, respectively; [math]\normalsize{\sigma^{x}}[/math], [math]\normalsize{\sigma^{y}}[/math] and [math]\normalsize{\sigma^{z}}[/math] are __ Pauli Matrices__, and [math]\normalsize{\gamma}[/math] 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__

**Figure 1**. A Pictorial Representation of a Wormhole Interior

**Figure 2**. Illustration of the Hologram for a Sphere

## 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**.

- Springer. "Euclidean Path Integrals"
__https://link.springer.com/chapter/10.1007/978-1-4612-0009-3_14__ - Inspire HEP. "Factorization and Non-Factorization of In-Medium Four-Quark Condensates"
__https://inspirehep.net/literature/676262__ - arXiv.org. "Editorial: New frontiers in holographic duality"
__https://arxiv.org/abs/2210.03315__ - Big Think. "Are we living in a baby universe that looks like a black hole to outsiders?"
__https://bigthink.com/hard-science/baby-universes-black-holes-dark-matter/__ - Wiley Online Library. "Energy-Efficient Memristive Euclidean Distance Engine for
Brain-Inspired Competitive Learning"
__https://onlinelibrary.wiley.com/doi/full/10.1002/aisy.202100114__ - Nature. "Strongly enhanced effects of Lorentz symmetry violation in entangled Yb+ ions"
__https://www.nature.com/articles/nphys3610__ - ResearchForLife7 (revisited from arXiv). "Discussion on Consistent Truncations: Uplifting the GPPZ Solutions"
__https://httpsresearchforlife7.com/wp-content/uploads/2024/05/Discussion_on_Consistent_Truncations__Uplifting_the_GPPZ_Solutions.pdf__ - Inspire HEP. "Fermion Zero Modes and Topological-charge on a
Domain Wall of the D-brane-like Dot"
__https://inspirehep.net/literature/1353725__