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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 Electronic Structure of atoms and molecules.

However, this method neglects Electron Correlation, which results from the inherent interactions between Electrons in quantum systems. The Correlation Energy [2], defined as the energy difference between the HF limit and the exact solution of the nonrelativistic Schrödinger Equation (Equation 1) (further information in here, Section 2), is one measure of this correlation.

[math]\LARGE{E_{c} = |E^{Exact} - E^{HF}| }[/math]

[math]\large{E_{c} = |E^{Exact} - E^{HF}| }[/math]

Equation 1. The Electronic Energy in HF Method

Nevertheless, other metrics, such as Statistical Correlation Coefficients [3] and Shannon Entropy, have been proposed to quantify electron correlation. Despite these methods, electron correlation remains challenging to calculate accurately for Large Systems [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 Operators (further information in here, 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 vacuum state. The general form of a two-electron state, [math]\small{| \Psi \rangle}[/math] can be written as:

[math]\LARGE{| \Psi \rangle = \sum_{a, b \in \{1, 2, \ldots, 2m\}} \omega_{a, b} c^\dagger_a c^\dagger_b |0 \rangle}[/math]

[math]\large{| \Psi \rangle = \sum_{a, b \in \{1, 2, \ldots, 2m\}} \omega_{a, b} c^\dagger_a c^\dagger_b |0 \rangle}[/math]

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 Von Neumann Entropy, like below.

[math]\LARGE{S(\rho) = - \operatorname{Tr}(\rho \log_2 \rho)}[/math]

[math]\large{S(\rho) = - \operatorname{Tr}(\rho \log_2 \rho)}[/math]

Equation 3. The Von Neumann Entropy equation

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

[math]\LARGE{\rho = \operatorname{Tr}|\Psi \rangle \langle \Psi|}[/math]

[math]\large{\rho = \operatorname{Tr}|\Psi \rangle \langle \Psi|}[/math]

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.

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

Figure 1. Schematic Representation of Electronic Structure of a Molecule

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 Hamiltonian for this system is given by:

[math]\LARGE{H = - \frac{J}{2} (1 + \gamma) \sigma^x_1 \sigma^x_2 - \frac{J}{2} (1 - \gamma) \sigma^y_1 \sigma^y_2 - B \sigma^z_1 \otimes I_2 - B I_1 \otimes \sigma^z_2}[/math]

[math]\large{H = - \frac{J}{2} (1 + \gamma) \sigma^x_1 \sigma^x_2 - \frac{J}{2} (1 - \gamma)}[/math]

[math]\large{\sigma^y_1 \sigma^y_2 - B \sigma^z_1 \otimes I_2 - B I_1 \otimes \sigma^z_2}[/math]

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

[math]\LARGE{S = \frac{1}{2} \log_2\left( \frac{1}{4} + \frac{1}{4 + {\lambda^2}} \right) + \frac{1}{\sqrt{4 + \lambda^{2}}} \log_2 \frac{( \sqrt{4 + \lambda^{2}} - 2} {\sqrt{4 + \lambda^{2}} + 2}}[/math]

[math]\large{S = \frac{1}{2} \log_2\left( \frac{1}{4 + {\lambda^2}} + \frac{1}{4 + {\lambda^2}}\right) +}[/math]

[math]\large{+ \frac{1}{\sqrt{4 + \lambda^{2}}} \log_2 \frac{( \sqrt{4 + \lambda^{2}} - 2} {\sqrt{4 + \lambda^{2}} + 2}}[/math]

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 here, Section 3).

Figure 3. Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group

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