Quantum entanglement in manganese(II) hexakisimidazole nitrate: on electronic structure imaging - A polarized neutron diffraction and DFT study

Quantum entanglement has been visualized for the first time, in view of the spin density distribution and electronic structure for manganese in manganese(II)hexakisimidazole nitrate. Using polarized neutron diffraction and density functional theory modelling we have found for the complex, which crystallizes in the R3̄ spacegroup, a = b = 12.4898(3) Å, c = 14.5526(4) Å, α = γ = 90°, β = 120°, Z = 3, that spatially antisymmetric and spatially symmetric shaped regions of negative spin density, in the spin density map for manganese, are a result of quantum entanglement of the high spin d5 configuration due to dative imidazole- manganese π- donation and σ-bonding interactions respectively. We have found leakage of the entangled states for manganese observed as regions of positive spin density with spherical (3.758(2) μB) and non-spherical (1.242(3) μB) contributions. Our results, which are supportive of Einstein's theory of general relativity, provide evidence for the existence of a black hole spin density distribution at the origin of an electronic structure and also address the paradoxical views of entanglement and quantum mechanics. We have also found the complex, which is an insulator, to be suitable for spintronic studies.


Introduction
In condensed matter, the state and structure of the electron is assessed through interactions with its environment. Quantum mechanics 1 suggests, for instance, for the entangled interactions between spin systems S A = ⎪i〉 A and S B = ⎪j〉 B in Hilbert space: that the final state of the system, ⎢ψ〉 AB , disentangles when either ⎪i〉 A or ⎪j〉 B is measured providing: where c ij, is the coefficient matrix in the subspace ij.
Otherwise, the final state of the system ⎢ψ〉 AB remains entangled (correlated) providing: The quantum mechanical interaction of a spin-½ electron with an applied field B z was first demonstrated in the Stern-Gerlach 2 experiment (figure 1, below).  figure 1, the doubly degenerate spatially antisymmetric precession cones about B z , for the quantum entangled spin states ⎢±½〉 of the spin-½ electron, confer an intrinsic C 3i point group. The C 3i point group breaks spherical (O h ) rotational symmetry about B z by lifting the t 2g (O h ) level and lowering the e g (O h ) level. The t 2g (O h ) is transformed according to the spatially antisymmetric doubly degenerate E g irreducible representation and the e g (O h ) transformed according to the spatially symmetric singly degenerate A g irreducible representation, of the C 3i point group. The doubly degenerate spatially antisymmetric precession cones about B z , for the quantum entangled spin states ⎢±½〉, suggest spin delocalization in the transformed t 2g (O h ) associated with the E g irreducible representation. By contrast, since there is no spatially symmetric precession about B z a vacuum (spherical solution) is suggested for the transformed e g (O h ) associated with the singly degenerate A g irreducible representation. We propose these qualitative symmetry assessments for the spin-½ electron in an applied field B z reflect the intrinsic point group of the electron with respect to its magnetic moment. a.
b. Figure 2: proposed schematic decomposing classical spherical symmetry for a spin-½ electron into irreducible states; a doubly degenerate spatially antisymmetric state (green-colored triangles), and a singly degenerate spatially symmetric state (black-lined sphere). Note: in b.) S z is arbitrarily chosen as the spin angular momentum quantization vector. Given figure 2b, we suggest that the spin angular momentum operator S for a spin-½ electron with an intrinsic electric field E be described as: where E 0 = the permittivity of vacuum in free space and i = cartesian components. Also, from figure 2b and equation (2), the electron's spin angular momentum arises from intrinsic spin precession (spin delocalization) in three-dimensional space.
We propose, according to figure 2b, classical spin entanglement for an electronic structure in Hilbert space, equations (3a -b) below. We describe a spatially antisymmetric classical entangled ferromagnetic spin exchange state, in the two-particle state problem: which is fermion-like, where n = 3 for three-fold orbital symmetry. We also describe a spatially symmetric classical entangled antiferromagnetic spin exchange state, which is boson-like: where, n = 1 for spherical orbital symmetry. The simultaneous occurrence of both fermion-like and boson-like spatial states in equations (3a and 3b), for our proposed description of an electronic structure in Hilbert space, is a likely consequence of an anyon-like 3 intermediate during the twoparticle spin states interchange. Further, since the electron has a rest mass (m e ) and rest energy it intrinsically follows E e = m e c 2 in view of Einstein's mass-energy 4 equivalence. In this respect, the boson-like state in equation (3b) and the fermion-like state in equation (3a) define the mass-like and light-like nature, respectively, of our proposed electronic structure in Hilbert space. Our conclusions, in this respect, are akin to an observer's (mass-like) view of "past" and "future" (light-like) cones in Minkowski 5 general relativistic space-time (figure 3, below).   The paradox of quantum formalism implies that a measurement in Planck's time, for example through the application of an applied field, on a member of a two-particle classical entangled spin exchange state collapses the classical system before any space-like information between entangled members can be transmitted. In this work we will consider Einstein's possible solution 6 to this paradox through our proposed view that hidden information, such as spatial symmetry (which confer spin distribution symmetry) and energy-momentum (which confer spin delocalization), pertaining to the classical entangled spin exchange state in Hilbert space, is carried by each classical entangled member and no information is transmitted between members during a quantum measurement. In this respect, we suggest the system remains entangled in the quantum world, albeit in a quantum entangled spin exchange state in Hilbert space (figure 4 below), during a quantum measurement on a member of the classical entangled world. Also, spatial and energy-momentum information leaked in the quantum entangled world will be representative of the hidden information of the classical entangled world. Furthermore, since we have proposed the coexistence of spatially antisymmetric and spatially symmetric classical entangled spin exchange states (equations (3a -b)) to be in view of the light-like and mass-like nature respectively of our proposed electronic structure in Hilbert space then their simultaneous assessment during measurement will be expected. Finally, we assume 'action at a distance' 6 (that is, entanglement which is implicit of electron correlation in our discussion) to be relative. We note the premise of Einstein's theory of general relativity 7 that gravity is an intrinsic geometric property of space-time and that the curvature of spacetime is related to the energy, momentum of matter present is in support of our proposed electronic structure in Hilbert space (equations (3a -b), figure 2b). For instance, since we have suggested no spin angular momentum is associated with the classical spatially symmetric (boson-like) entangled state (equations (2, 3b), figure 2) then Minkowski 5 space-time (figure 3) for the associated spin density distribution, during a quantum measurement, is expected to be spherical and localized (at t = 0, the origin in figure 3). This spherical solution, referred to as the vacuum solution to Einstein's field equations (EFE) 8-9 since the energy-momentum tensor in EFE [8][9] is zero, is defined by the Schwarzschild metric [10][11] . Further, a black hole 12 , an object with a radius smaller than the Schwarzschild radius 10-11 and having a predicted singularity [13][14] , is also expected to be found. (This suggests, within the context of the associated spin density distribution, a 'black hole spin density distribution' at the origin). By contrast if the energy-momentum tensor in EFE [8][9] is not zero Minkowski 5 space-time (figure 3) is spatially antisymmetric (at t = +t and t = -t in figure 3). In this respect, we expect during measurement that the spin density distribution associated with our proposed classical spatially antisymmetric (fermion-like) entangled state, which has intrinsic orbital angular momentum (equations (2, 3a), figure 2b), to be delocalized and spatially antisymmetric having C 3i symmetry.

Experimental overview
To explore experimentally and theoretically the aforementioned considerations, we investigate the spin density distribution and electronic structure for the interactions of manganese in the titled complex (figure 5 below) using polarised neutron diffraction (PND) 15 and density functional theory modeling.

Classical spin exchange entanglement, in Hilbert space, for the interactions of manganese and imidazole in the titled complex.
We propose 16 that if the high spin d 5 -configuration for Mn(II) has the α -spin direction, or the β-spin direction, an occupied Φ 3d α-spin orbital, or an occupied Φ 3d β-spin orbital, is defined. The Lewis acidity of Mn(II) in this context, with respect to a coordinating imidazole moiety, is viewed as an unoccupied Φ 3d spin orbital on Mn(II) in the presence of an occupied spin orbital associated with a coordinating pyridine-like nitrogen of imidazole (Figure 5b), where imidazole acts as a Lewis base. In this respect, while considering the valence bond (also Heitler-London model), localized bonding, independent bond approaches, [16][17][18][19][20][21] we suggest classical spin entanglement for an electronic structure in Hilbert space (equations (3a -b), figure 2b) associated with the interactions of manganese and imidazole in the titled complex. Within the context of spin states interaction, we suggest: • Dative imidazole-manganese σ -bonding interaction, involving the spin states (arbitrarily assigned α(↑) and β (↓)) for the pair of electrons in the available sp 2 -hybridized orbital of the pyridine-like nitrogen of imidazole interacting with an unoccupied Φ 3d spin orbital associated with the e g set of Mn(II). Additionally, we propose 16 that dative imidazole-manganese σbonding interactions constitute antiferromagnetic spin exchange interaction, in the two particle problem, such that a classical spatially symmetric entangled exchange state in Hilbert space is defined as shown in equation (3b).
• Dative imidazole-manganese π-donation interaction, involving the unpaired electron in the p z orbital of the pyridine-like nitrogen of the coordinating imidazole π-donating unto an unoccupied Φ 3d spin orbital associated with the t 2g set of Mn(II). This constitutes ferromagnetic spin exchange interaction, in the two particle state problem, such that a classical spatially antisymmetric entangled exchange state in Hilbert space is conferred equation (3a)).

Leakage of hidden information (spatial symmetry, energy-momentum) during a PND measurement of the titled complex:
• For dative imidazole-manganese σ -bonding interaction we suggest the leaked spatial information during measurement, associated with our proposed classical spatially symmetric entangled state in Hilbert space (equation (3b)), be described as the spherical symmetric 6 A 1g (t 2g 3 e g 2 ) high spin crystal field state for manganese associated with the σ* orbital on manganese. Also, since no orbital angular momentum is associated with dative imidazolemanganese σ-bonding interactions 16 , as the σ-bond quantization axis taken as the z-direction is projected towards the manganese nucleus, no spin delocalization (that is, zero energymomentum) is associated with the 6 A 1g (t 2g 3 e g 2 ) state. Further, in accordance with the theory of general relativity 7 , a zero value for the energy-momentum tensor in EFE 8-9 imparts a vacuum solution to EFE [8][9] . This also confers an implicit black hole 12 having a radius smaller than the Schwarzschild radius 9-10 . We therefore expect an imaged black hole spin density distribution for our proposed 6 A 1g (t 2g 3 e g 2 ) state for manganese during measurement. Analytically we define the measured magnetic moment µσ*(Mn), associated with the hole in the σ* orbital on manganese, as: with S = 5/2, n 1 = 1 for the spherical 6 A 1g (t 2g 3 e g 2 ) crystal field state, k Mn,σ is the admixture coefficient on manganese (assessed from DFT modeling) due to dative imidazole-manganese σ-bonding interaction, g = 2 for the electron g-factor, µ B is the Bohr magneton.
• For dative imidazole-manganese π-donation interaction we suggest the leaked spatial information during measurement, associated with our proposed classical spatially antisymmetric entangled state in Hilbert space (equation (3a)), be described as the three-fold spatially antisymmetric 2 T 1g (t 2g 5 e g 0 ) low spin crystal field state for manganese associated with the π* orbital on manganese. Also, since orbital angular momentum is intrinsic to dative imidazole-manganese π-donation interactions 16 , as the π-donation quantization axis taken as the z-direction is projected away from the manganese nucleus, spin delocalization (energymomentum) is associated with the 2 T 1g (t 2g 5 e g 0 ) state. We define the measured magnetic moment µπ* (Mn), associated with the π* orbital on manganese, as: µ π* (Mn) = (1 -(n 2 η Mn + n 2 k Mn, π ))g µ B S with S = 5/2, n 2 = 3 for the three-fold 2 T 1g crystal field state, k Mn,π is the admixture coefficient on manganese (assessed from DFT modeling) due to dative imidazole-manganese π-donation interactions, η M is the spin delocalization for manganese.
Since π-donation by the coordinated imidazoles unto the t 2g set of manganese breaks the aromaticity of the imidazoles then spin delocalization η Imz(Mn) for manganese (associated with the 2 T 1g state) is expected to be observed in the π* orbital of each imidazole. We also expect spin delocalization η Imz(Mn) for manganese to be observed in the σ* orbital of each imidazole.
2.3 Given the spin density form factor f S in q-space: where, ρ is the spin density and f S (q) is the fourier transform: then, in terms of multipolar expansion 22 , the spin density form factor is re-expressed as: where, l = 0, 2, 4 and -l≤ m≤ l, 〈j n,l 〉 is the radial integral and, Z

PND results
Re-construction of the spin density map for manganese (Figure 6a, below), by inverse Fourier 23 of the observed spin density structure factors obtained from the PND experiment on a single crystal of the complex, shows regions of negative and positive spin density. The regions of negative spin density (see figure 6a) are due to spatially antisymmetric quantum entangled spin exchange (*) and spatially symmetric quantum entangled spin exchange (**) states, in Hilbert space, for manganese. We note the quantum entangled spin exchange states for manganese are associated with classical entangled spin exchange states of our proposed electronic structure for the complex in Hilbert space (equations (3ab), figure 2, section 2.1). In figure 6a, the region of positive spin density at the origin, which was found to have a value 3.758(2)µ B using Multipolar Least Square Analysis (MPLSQ) 23 , is due to leakage of the spherical symmetric 6 A 1g (t 2g 3 e g 2 ) crystal field state for manganese associated with the spatially symmetric classical entangled state (sections 2.1 -2.2). As shown in figure 6a, the spin density distribution at the origin constitutes a black hole spin density distribution for manganese. This is because no spin (electron) delocalization (that is, no energy-momentum) is associated with the 6 A 1g (t 2g 3 e g 2 ) state for manganese. Our observation of a black hole spin density distribution at the origin (see also our discussion at the end of section 1) for manganese in figure 6a is supported by the theory of general relativity 7 , which predicts the occurrence of a black hole as an intrinsic consequence of the vacuum solution to EFE [8][9] . This conveys with relevance to dative imidazole-manganese σ-bonding interactions in the titled complex, a collapsing of the mass-like nature of spin density distribution for manganese at the origin, of our proposed electronic structure in Hilbert space (sections 2.1 -2.2, equations (3a -b), figure 2b). The Schwarzschild radius 10-11 for S = 5/2 was calculated to be 0.6765 × 10 -54 cm. We have measured a Schwarzschild radius of 0.65 cm based on figure 6a. The factor of 10 -54 between the calculated and measured Schwarzschild radius is attributed to scaling.  In figure 6a the regions of positive spin density projected away (green-colored contours, figure 6a) from the origin, which were assessed to be 1.242(3) µ B , are a result of spin delocalization (energy-momentum) for manganese. This is due to our proposed leakage of a (non-spherical) threefold spatially antisymmetric 2 T 1g (t 2g 5 e g 0 ) low spin crystal field state for manganese. Spin delocalization of the 2 T 1g (t 2g 5 e g 0 ) state for manganese, which is associated with the π* orbital on manganese, occurs also in the π* and σ* orbitals of the coordinated imidazole. In this respect our assessed value of 1.242µ B for the regions of positive spin density projected away from the origin in figure 6a takes in account equation 6 of section 2.2. Lastly, we conclude the value 1.242µ B /5µ B = ¼, which is indicative of the entropy associated with a black hole 24 , is a result of orbital angular momentum conferring non-spherical spin density distribution for manganese. This inherently reduces the mass (in Bohr magneton units) of the black hole spin density distribution for manganese at the origin (in figure 6a) from 5µ B to 3.758µ B .

DFT modeling overview, results
In modeling the spin density for manganese in the titled complex, the Dmol3 package 25-26 was employed. We note the spin density ρ σ (r), given as a sum over all occupation points (k) for the function (Φ k, σ): also, in spin basis σ, the density is represented: ρ = ρ σ + ρ -σ and, due to variations in density, the total energy E t minimized is given as: where, V e defines the electrostatic potential, α and β are spin up and spin down states respectively, and µ xc the exchange correlation. A general eigenvalue matrix equation, [25][26] where the matrix elements in the subspace ij are given as: Dmol3 25-26 uses numeric orbitals φ i (r) as the basis set calculated from multipolar functions. The multipole functions, which take into account the point group symmetry of the complex are defined through the expansion: The results of the DFT modeled spin density on manganese (figure 6c) intrinsically represents a leakage of spin density information consequential of the C 3i point group for the complex used to determine the numeric input basis set in equation (11). Comparing the Mulliken spin population plot (figure 6b) with the spin density plot (figure 6c), we note the spatially antisymmetric and spatially symmetric shaped regions of negative spin density in figure 6c are associated with states (i) and (ii) respectively in figure 6b. These are due to off-diagonal matrix elements for equation (10). The diagonal matrix elements for equation (10), attributed to triangular and circular contour regions of positive spin density along the C 3i axis in figure 6c, are associated with states (iii) and (iv) in figure 6b. States (iii) and (iv) in figure 6b, which are doubly and singly degenerate respectively, are contributions arising from the transformation of the t 2g (O h ) and e g (O h ) sets for manganese according to the doubly degenerate E g and singly degenerate A g irreducible representations respectively of the C 3i point group. While states (i) and (ii) in figure 6b, which are doubly and singly degenerate respectively, are contributions arising from the transformations of s-type and p-type orbitals (for the coordinating pyridine-like nitrogen of imidazole) according to the doubly degenerate E u and singly degenerate A u irreducible representations respectively of the C 3i point group.
In comparing the DFT modeled spin density plot (figure 6c) and the PND spin density plot (figure 6a) the black hole spin density distribution for manganese at the origin in the PND spin density plot is absent in the DFT modeled spin density plot. This difference, we propose, is due to general relativistic 7 considerations of energy-momentum, not taken into account in the DFT modeling which assumes special relativisty 27 . Furthermore, the DFT spin density model assumes complete decoupling of the entangled states shown in figure 7 below. Figure 7: DFT modeled entangled states for the complex associated with π* and σ* orbitals on manganese due to dative imidazole-manganese π-donation interactions and dative imidazolemanganese σ-bonding interactions, respectively.
Given the results of the Mulliken spin population plot in figure 6b the admixture co-efficient for manganese k Mn,π (equation (5)) associated with dative imidazole-manganese π-donation interactions (figure 7a) was assessed to be 0.08. While the admixture coefficient for manganese k Mn,σ (equation (4)) associated with dative imidazole-manganese σ-bonding interactions (figure 7b) was assessed to be 0.25. Given the general relativistic 7 consideration of energy-momentum, which is in view of spin delocalization η Mn (equation (5)) for manganese associated with dative imidazolemanganese π-donation interactions, the value of the admixture co-efficient for manganese k Mn,π was re-assessed to be 0.29 (η Mn + k Mn,π , with η Mn = 0.21).

Electronic structure, suitability of the complex for spintronic studies.
It is well known that quantum entanglement, an implicit consequence of bonding interactions between two spin systems, is associated with the electronic structure arising from antiferromagnetic (spin singlet) exchange and ferromagnetic (spin triplet) exchange in Hilbert space. The antiferromagnetic spin exchange is spatially symmetric and the ferromagnetic spin exchange is spatially antisymmetric. If the exchange interaction is strongly ferromagnetic the spin triplet exchange state is lower in energy than the spin singlet exchange state. Otherwise, if the exchange interaction is strongly antiferromagnetic the spin singlet exchange state is higher in energy than the spin triplet exchange state. Given the results for the titled complex, we suggest the spin triplet exchange state is associated with a non-spherical spin density distribution and the spin singlet exchange state associated with a spherical spin distribution. Using the nuclear structure for the titled complex, we modeled complex stability (figure 8, below) in view of trigonal (non-spherical) stability for the series [M(Imidazole) 6 ] 2+ (NO 3 -) 2 , where M is a 3d-transition metal ion. According to figure 8, complex stability is greater for the d 5 configuration. The titled complex is therefore the best candidate in the series for observing quantum entangled, electronic structure in Hilbert space. The suitability of the titled complex for spintronic studies, such as allowing the flow of spinpolarized currents over its surface, is firstly due to its expected conducting nature (state (iv) (figure 6b), which is spatially singly degenerate, being close to the Fermi level). And, secondly, due to the complex's expected insulating nature (state (iii) (figure 6b), which is spatially doubly degenerate, being further below the Fermi level in the bulk). The black hole spin density distribution for manganese at the origin in the PND spin density map (figure 6a), which we associate with the spherical symmetric 6 A 1g (t 2g 3 e g 2 ) state for manganese, indicate the ability of the titled complex to be spin polarizing at its surface (see also state (iv), figure 6b) to spin polarized currents. While the spin density distribution for manganese projected away from the origin in the PND spin density map (figure 6a), which we attribute to spin delocalization associated with the three-fold spatially antisymmetric 2 T 1g (t 2g 5 e g 0 ) state for manganese, indicate the ability of the titled complex to be spin depolarizing, insulating, in its bulk (see also state (iii), figure 6b) to spin polarized currents.

Conclusion
We have shown for the first time using polarized neutron diffraction, density functional theory a visualization of quantum entanglement in view of the spin density distribution, electronic structure associated with the interactions of manganese in the titled complex. Our results for the titled complex is in support of the Einstein's theory of general relativity and show that the solution to the paradox of quantum formalism assumes leakage of hidden information pertaining to the entangled state during measurement. Finally, our results show the suitability of the complex for spintronic studies. Figure C1: Magnetization measurement on a powdered sample of the titled complex at 2 K , showing saturation at 5µ B . The frequency independent splitting in the spectra cannot be described as an effect of electron gfactor anisotropy.