Far down the Quantum Chemistry: how to Intoduce an “Entangled” Electron Correlation


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 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 (Equation 1) of the nonrelativistic Schrödinger Equation (further information in here, Section 2), is one measure of this correlation.

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

[math]\normalsize{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 space, represented by fermionic Annihilation and Creation Operators (further information in here, Section 2), [math]\normalsize{c_{a}}[/math] and [math]\normalsize{{c^{\dagger}}_{a}}[/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]\normalsize{| \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, 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]\normalsize{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]\normalsize{\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.

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]\small{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]

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 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]\normalsize{S = \frac{1}{2} \log_2\left( \frac{1}{4 + {\lambda^2}} + \frac{1}{4} + k^2 \right) + \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 1.   A Pictorial Representation of a Wormhole Interior

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

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.

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  2. Inspire HEP. "Factorization and Non-Factorization of In-Medium Four-Quark Condensates" https://inspirehep.net/literature/676262

  3. arXiv.org. "Editorial: New frontiers in holographic duality" https://arxiv.org/abs/2210.03315

  4. 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/

  5. Wiley Online Library. "Energy-Efficient Memristive Euclidean Distance Engine for Brain-Inspired Competitive Learning" https://onlinelibrary.wiley.com/doi/full/10.1002/aisy.202100114

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  8. Inspire HEP. "Fermion Zero Modes and Topological-charge on a Domain Wall of the D-brane-like Dot" https://inspirehep.net/literature/1353725


The Relation between the Uncertainty Principle and the Gravitation within the Irreconcilable Theories Enigma

The First Steps towards an Overall Vision

One of the persistent and demanding challenges in the realm of Physics is grappling with the inquiry of whether the Gravitational Force exhibits inherent quantum characteristics and, if so, how to construct a comprehensive quantum framework for Gravity that sidesteps conceptual dilemmas while maintaining predictive efficacy across all Energy scales. As the pursuit of amalgamating gravitational and quantum phenomena advances, it prompts the consideration of whether the foundational tenets of Quantum Mechanics necessitate reevaluation within the domain of Quantum Gravity (further information in here, section 1).

The integration of quantum and gravitational effects within a unified framework introduces nuanced complexities. Numerous quantum gravity models postulate a minimum length scale at the Planck Scale, indicating a fundamental limitation in the resolution of Space - Time. This Planck Length, lp serves as a natural threshold beyond which Space - Time is hypothesized to exhibit a granular, foamy structure due to inherent quantum fluctuations. Consequently, several studies advocate for the modification of the Heisenberg Uncertainty Principle (further information in here, section 2) at the quantum gravity scale to accommodate this fundamental length.

Extending the Uncertainty Concept...

It is widely recognized that a cornerstone of quantum mechanics resides in the Heisenberg Uncertainty Principle (HUP). However, it’s important to note that there isn’t a predetermined quantum limit on the precision of individual position or linear momentum measurements; theoretically, arbitrarily short distances can be probed using exceedingly high energy probes, and conversely. One prevalent generalization, known as the Generalized Uncertainty Principle (GUP), is expressed as:

[math]\LARGE{\delta x \delta p \geq \frac{\hbar}{2} \pm 2|\beta|^2 {{l_{p}}^2} \frac{\delta p}{\hbar} = \frac{\hbar}{2} \pm 2|\beta| \hbar \frac{ \delta p}{{m_{p}}^2}}[/math]

[math]\normalsize{\delta x \delta p \geq \frac{\hbar}{2} \pm 2|\beta|^2 {{l_{p}}^2} \frac{\delta p}{\hbar} = \frac{\hbar}{2} \pm 2|\beta| \hbar \frac{\delta p}{{m_{p}}^2}}[/math]

Equation 1.   The GUP Equation

where [math]\normalsize{\delta{x}}[/math] and [math]\normalsize{\delta{p}}[/math] are the position and linear momentum uncertainties, respectively; [math]\normalsize{\hbar}[/math] is the Planck constant and [math]\normalsize{m_{p}}[/math] is the mass of particle.

Here, the sign [math]\Large{\pm}[/math] denotes positive or negative values of the dimensionless Deformation Parameter, [math]\Large{\beta}[/math] [1] typically assumed to be of order unity in certain quantum gravity models, such as String Theory (further information in here, section 1). However, alternative derivations and experimental inquiries scrutinize the phenomenological implications of this fundamental parameter. The consideration that it could near zero leads to the recovery of standard quantum mechanics, implying that modifications to the HUP become significant only at the Planck scale. Furthermore, for mirror-symmetric states [2] (i.e. [math]\Large{\hat{p} = 0}[/math]), a one can derive the following modified commutator relation.

[math]\LARGE{[\hat{x}, \hat{p}] = i\hbar \left( 1 \pm |\beta| \Biggl(\frac{\hat{p}^2}{{m_{p}}^2} \right)\Biggl)}[/math]

[math]\large{[\hat{x}, \hat{p}] = i\hbar \left( 1 \pm |\beta| \Biggl(\frac{\hat{p}^2}{{m_{p}}^2} \right)\Biggr)}[/math]

Equation 2.   Modified Generalized Commutation Relation expression

[math]\Large{\hat{x}}[/math] and [math]\Large{\hat{p}}[/math] are, in the order, the position and linear momentum operators.

While the assumption of [math]\normalsize{\beta}[/math] being of order unity enjoys widespread acceptance and empirical support from various contexts beyond String Theory, the debate over the sign persists. Arguments advocating for a negative [math]\normalsize{\beta}[/math] suggest compatibility with scenarios featuring a lattice-like structure underlying the Universe, or alignment with observational constraints like the Chandrasekhar Limit [3] for White Dwarfs.

Figure 1.   write

write (Figure 3) write,

Has a Black Hole a Quantum Aspect?

Corpuscular Gravity (CG), offers an alternative framework, describing Black Holes as Bose-Einstein Condensates [4] of Gravitons at the critical point of a Quantum Phase Transition [5]. By linking GUP black hole thermodynamics with corpuscular gravity, researchers aim to reconcile these two disparate theories. To do so, let’s examine the GUP-modified expressions of the Emission Rate of Black Holes, expanded up to the order O ([math]\normalsize{1/M^4}[/math]).

[math]\LARGE{\Biggl(\frac{dM}{dt}\Biggr)_{{\text{GUP}}} = -\frac{1}{60(16)^2 \pi} \Biggl(\frac{m^4_p}{\hbar M^2} \pm |\beta| \frac{m^6_p}{\hbar M^4}\Biggr)}[/math]

[math]\normalsize{\Biggl(\frac{dM}{dt}\Biggr)_{{\text{GUP}}} = -\frac{1}{60(16)^2 \pi} \Biggl(\frac{m^4_p}{\hbar M^2} \pm |\beta| \frac{m^6_p}{\hbar M^4}\Biggr)}[/math]

Equation 3.   The GUP Emission Rate equation

While, for CG theory:

[math]\LARGE{\Biggl(\frac{dM}{dt}\Biggr)_{{\text{CG}}} = -\frac{1}{60(16)^2 \pi} \Biggl[\frac{m^4_p}{\hbar M^2} +}[/math] O [math]\LARGE{\Biggl(\frac{m^6_p}{\hbar M^4}\Biggr)}\Biggr][/math]

[math]\normalsize{\Biggl(\frac{dM}{dt}\Biggr)_{{\text{CG}}} = -\frac{1}{60(16)^2 \pi} \Biggl[\frac{m^4_p}{\hbar M^2} +}[/math] O [math]\normalsize{\Biggl(\frac{m^6_p}{\hbar M^4}\Biggr)}\Biggr][/math]

Equation 4.   The CG Emission Rate equation

in which [math]\small{M}[/math] is the Black Hole Mass; [math]\small{t}[/math] is the time coordinate.

As said before, and specifically in the case of Black Holes, the deformation parameter can be positive or negative valued. However, it's possible to prove, at least up to the first order, that the corrections induced by these two theories exhibit the same functional dependence on the black hole mass.

Again, since the coefficient in front of the correction is predicted to be of order unity, numerical consistency between the GUP and CG expressions automatically leads to
[math]\normalsize{\beta \sim}[/math] ([math]1[/math]), which is in agreement with predictions of other models of quantum gravity. Therefore, despite their completely different underlying backgrounds, the GUP and CG approaches are found to be compatible with each other.

Figure 2.   write write

Figure 3.   write

The Importance of Deformation Parameter in the Unification of Quantum Theory and Gravity

Further research is needed to elucidate the precise nature of the GUP deformation parameter, and its implications for the behavior of black holes and quantum gravity. Exploring alternative scenarios, such as [math]\normalsize{\beta}[/math] as a function rather than a constant, promises to deepen our understanding of the intricate interplay between gravity and quantum mechanics.

  1. ResearchForLife7 (revisited from IOPscience). "A Discussion on Deformation Parameter Features"https://httpsresearchforlife7.com/wp-content/uploads/2024/04/A-Discussion-on-Deformation-Parameter-Features.pdf

  2. National Institutes of Health (NIH). "Mirror simmetry breaking at the molecular level" https://www.ncbi.nlm.nih.gov/pmc/articles/PMC38075/pdf/pnas01525-0160.pdf

  3. Space.com. "The Chandrasekhar limit: Why only some stars become supernovas" https://www.space.com/chandrasekhar-limit#:~:text=What%20is%20the%20Chandrasekhar%20limit,the%20mass%20of%20the%20sun.

  4. ScienceDaily. "First quasiparticle Bose-Einstein condensate" https://www.sciencedaily.com/releases/2022/10/221025120127.htm

  5. Quantamagazine. "Physicists Observe ‘Unobservable’ Quantum Phase Transition" https://www.quantamagazine.org/physicists-observe-unobservable-quantum-phase-transition-20230911/


Is it Really Possible to Overcome the Wave-Particle Duality?

Undoubtedly, at the dawn of Quantum Theory birth there is the comparison between the David Bohm's Deterministic Interpretation, with hidden variables, and the Schrödinger and Born Probabilistic Paradigm, which introduces the Wavefunction, 𝜓 as a representation of a physical system's state (Figure 1).

Towards a Unified Quantum Mechanics Theory

In the last years, a Unified Quantum Mechanics framework has been proposed, consisting of a comprehensive analysis of the WaveParticle Dynamics of quantum particles. The conventional wavefunction, [math]\Large{\psi}[/math] (Figure 2) is redefined in terms of real scalar [math]\Large{{R}_0}[/math] and vector [math]\Large{\vec{R}}[/math] functions, shedding light on the intrinsic nature of quantum particles. The resulting expression unveils a dual composition of massless and massive fields within the particle, with the scalar [math]\Large{{R}_0}[/math] and vector [math]\Large{\vec{R}}[/math] functions satisfying the Quantum Telegraph Equation (Equation 1) [1] and the phase, [math]\Large{S}[/math] representing a massless field.

The interaction between these three functions is investigated, revealing intriguing connections. Notably, the product
[math]\Large{S \cdot {R_0}}[/math] satisfies the telegraph equation, representing the potential energy of the particle field, while [math]\Large{S \cdot {\vec{R}}}[/math] signifies its momentum. This article explores scenarios where the particle exhibits fluid-like (wave) behavior, characterized by a conserved Energy–Momentum Tensor, [math]\Large{{\sigma_{ij}}^m}[/math] (Equation 2).

[math]\LARGE{ - {\tau\hbar}{{\partial^2\psi} \over {\partial t^2}} + {i\hbar} {{\partial\psi} \over {\partial t}} = - {{\hbar^2} \over 2m} \Delta\psi}[/math]

[math]\Large{ - {\tau\hbar}{{\partial^2\psi} \over {\partial t^2}} + {i\hbar} {{\partial\psi} \over {\partial t}} = - {{\hbar^2} \over 2m} \Delta\psi}[/math]

Equation 1.   The mathematical general form of the Quantum Telegraph Equation

  • [math]\Large{\tau}[/math]          is a Time-dimension Parameter;

  • [math]\Large{\hbar}[/math]          is the Planck Constant;

  • [math]\Large{\psi}[/math]         is the three-dimensional Electron Wavefunction;

  • [math]\Large{t}[/math]           is the Time coordinate;

  • [math]\Large{i}[/math]           is the Imaginary Number;

  • [math]\Large{m}[/math]       is the Electron Mass;

  • [math]\Large{\Delta{\psi}}[/math]   is the Electron Wavefunction three-dimensional Espansion;

  • [math]\Large{v_p}[/math]       is the Phase Velocity;

  • [math]\Large{p^{\infty}}[/math]    is the Phase at high-frequency ([math]\Large{\omega}[/math]) limit;

Furthermore, Bohmian Mechanics can be incorporated into the Dirac Equation, resulting in a relativistic Hamilton – Jacobi Equation and a Continuity Equation. The new formalism eliminates the conventional quantum potential energy, providing a fresh perspective on Quantum Mechanics. Application of the Bohmian Method to Maxwell's Equations demonstrates the Particle–Wave Duality [2] of the electromagnetic field. It can also be established that when the electromagnetic field behaves as matter, it resembles a massless Dirac particle, supporting the de Broglie Hypothesis [3]. 

[math]\LARGE{{{\sigma_{ij}}^{m}} = { - {\vec{R}_i}{\vec{R}_j} + {\delta_{ij}}{\Biggr({{\vec{R}^2} \over {2}} - {{\vec{R}_0}^2 \over {2{c_{0}}^2}}\Biggl)}}}[/math]

[math]\large{{{\sigma_{ij}}^{m}} = { - {\vec{R}_i}{\vec{R}_j} + {\delta_{ij}}{\Biggr({{\vec{R}^2} \over {2}} - {{\vec{R}_0}^2 \over {2{c_{0}}^2}}\Biggl)}}}[/math]

Equation 2.   The Energy-Momentum Tensor mathematical expression

where [math]\vec{R}_i[/math] and [math]\vec{R}_j[/math] are the [math]R[/math] vector components along the [math]i[/math] and [math]j[/math] arbitrary directions, respectively; while [math]\delta_{ij}[/math] is the Delta Function, which modulates the tensor magnitude.

What about a New Potential Energy?

In the so-called Schrödinger-Bohm Mechanics [4], the motion of a non-relativistic quantum particle in a Potential Energy, [math]\Large{V}[/math] is described using the Schrödinger Equation. The wavefunction is expressed as:

[math]\LARGE{\psi = {Re^{{iS} \over {\hbar}}}}[/math]

[math]\Large{\psi = {Re^{{iS} \over {\hbar}}}}[/math]

Equation 3.   One of the Electron Wavefunction scalar forms

where [math]\Large{R}[/math] and [math]\Large{S}[/math] are real functions. The resulting equations are analyzed, revealing the dual nature of the particle as both wave and matter. A new potential energy dependent on the phase [math]\Large{S}[/math] is introduced, termed the Phase (Spin) Potential Energy.

A hybrid Theory of Vector and Scalar Wavefunctions

The Vector Quantum Mechanics represents a free quantum particle with mass, [math]\Large{m}[/math] using scalar and vector wavefunctions [math]\Large{\psi{_0}}[/math] and [math]\Large{\vec{\psi}}[/math], respectively. The quantum telegraph equation governs their evolution, illustrating the fluid-like motion [5] of the particle.


Figure 1.   A representative 3-Dimensional Electron Wavefunction Energy Distribution graph

From Bohmian Vector Quantum Mechanics to Relativistic Dirac Equation

The combination of Bohmian Vector Quantum Mechanics to vector quantum mechanics yields to a relativistic Dirac Hamiltonian for massless and massive particles. The Hamiltonian equality implies a dual description of the particle, with all information propagating at the speed of light. The mass of the particle is uniquely determined by its wavefunctions, showcasing the interconnectedness of the fields.

Application to Maxwell’s Equations

The Bohmian approach is further extended to Maxwell's equations, revealing the particle–wave duality of the Electromagnetic Field. The electromagnetic field is expressed as a wave and particle, with the phase [math]\Large{S}[/math] governing the particle dynamics and [math]\Large{\vec{E}_0}[/math] and [math]\Large{\vec{B}_0}[/math] (Electric Field and Magnetic Field in vacuum, respectively, interacting with particle) representing wave aspects. The Photon, when behaving as matter, resembles a massless Dirac particle, supporting the de Broglie hypothesis.


Figure 2.   A fountain plan of Electron Wavefunctions Interference

A Delicious Chance to Connect the Matter and Wave features of Particles

The Unified Quantum Mechanics framework highlights the inherent duality and interconnectedness of massless and massive fields within quantum particles. Particularly, the elimination of the quantum potential energy in this new formalism opens avenues for further exploration in quantum theory.

  1. LinkedIn. "Classical and Quantum Wave Equations: A Journey into the Core Concepts"https://www.linkedin.com/pulse/classical-quantum-wave-equations-journey-core-concepts-dadhich/

  2. ScienceDirect.com. "Complementarity, wave-particle duality, and domains of applicability" https://www.sciencedirect.com/science/article/pii/S1355219817301028

  3. ScienceReady.com. "de Broglie's Matter Wave Duality and Experimental Evidence" https://scienceready.com.au/pages/matter-wave-duality

  4. IOPscience. "Is the de Broglie-Bohm interpretation of quantum mechanics really plausible?" https://iopscience.iop.org/article/10.1088/1742-6596/442/1/012060/pdf

  5. Springer.com. "Visualization of hydrodynamic pilot-wave phenomena" https://link.springer.com/article/10.1007/s12650-016-0383-5

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