On bicriticality of (3,6)-fullerene graphs

A (3,6)-fullerene is a plane cubic graph whose faces are only triangles and hexagons. A connected graph G with at least \(2n+2\) vertices is said to be n-extendable if it has n independent edges and every set of n independent edges extends to a perfect matching of G. A graph G is said to be bicritical if for every pair of distinct vertices u and v, \(G-u-v\) has a perfect matching. It is known that every (3,6)-fullerene is 1-extendable, but not 2-extendable. In this short paper, we show that a (3,6)-fullerene G is bicritical if and only if G has the connectivity 3 and is not isomorphic to one graph (2,4,2). As a surprising consequence we have that a (3,6)-fullerene is bicritical if and only if each hexagonal face is resonant.     

Structural features of films based on star-shaped fullerene-containing polystyrenes: Small-angle neutron-scattering study

The structuring of films based on regular six-arm and twelve-arm polystyrenes with the arm characteristics M_n = 118 × 10³ and M_w/M_n = 1.06 and a fullerene (С₆₀) branching center in a weakly swollen state in a mixture of deuterated solvents (90 vol % D-methanol and 10 vol % D-toluene) is studied by small-angle neutron scattering. Analogous studies are performed for films based on linear polystyrene (M_n = 280 × 10³, M_w/M_n = 1.07) and its composites with fullerene С₆₀ (0.5 wt %). It is shown that, during saturation of the samples with the solvent D-toluene, filling of the free volume that forms chain-transport channels that have gyration radii of the order of diameters of macromolecules and unite to form submicron and micron structures occurs. The degree of filling of the free volume, which is maximum for the linear polymer, becomes minimum for polystyrene with the inserted free fullerene. In addition, the transition from the linear polymer to the six-arm polymer is related to a smaller degree of filling of the free volume due to the formation of thinner migration channels. As the amount of arms in macromolecules is increased to 12, permeability of the polymer matrix improves because molecular order partially appears in accordance with the model of the densest packing of solid spheres.     

Supramolecular architecture of catechol and its 2:1 complex with dimethylsulfoxide

The crystal structures of catechol (o-dihydroxybenzene) and its 2:1 complex with dimethylsulfoxide are determined at T = 150 K. Crystal data: C₁₄H₁₈O₅S, M = 298.37, triclinic, space group P \(\bar 1\), unit cell parameters: a = 7.7285(13) Å, b = 9.9924(17) Å, c = 10.3188(18) Å, α = 89.963(4)°, β = 89.968(4)°, γ = 69.076(5)°, V = 744.3(2)ų, Z = 2, D _x = 1.331 g/cm³, R1 = 0.048; C₆H₆O₂, M = 110.11, monoclinic, space group P2₁/n, a = 9.8206(6)Å, b = 5.5903(3)Å, c = 10.4439(6)Å, β = 114.952(2)°; V = 519.85(5) ų, Z = 4, D _x = 1.407 g/cm³, R1 = 0.0289. In the 2:1 complex the molecules are joined in a supramolecular ensemble by D-H...A hydrogen bonds (D = O, C; A = O, π); in catechol they are bonded only by O-H...O. The state diagram of the catechol-dimethylsulfoxide system is examined by DTA.     

Synthesis of Block Copolymers of Styrene with <Emphasis Type="Italic">D</Emphasis>,<Emphasis Type="Italic">L</Emphasis>-Lactide by the Sequential Controlled Cationic Polymerization and Ring-Opening Anionic Polymerization

The cationic polymerization of styrene initiated by the system 2-chloro-2-phenylpropane–TiCl₄–pyridine is studied in a mixture CH₂Cl₂–n-hexane at a temperature of –80°С. It is shown that under these conditions polymerization occurs via the living mechanism at [monomer]: [initiator] ≤ 100. The method of preparing polystyrenes with terminal primary hydroxyl groups (M_n = 4000–10000 g/mol) by the sequential controlled cationic polymerization of styrene and the in situ alkylation of 4-phenoxy-1-butanol by polystyrene macrocations is proposed. The resulting functionalized polystyrenes are used as macroinitiators of anionic-coordination ring-opening polymerization of D,L-lactide in the presence of tin bis(2-ethyl hexanoate) [Sn(Oct)₂] in toluene at 80°С. Copolymers polystyrene-block-poly(D,L-lactide) with the controlled length of the poly(D,L-lactide) block (M_n = 10000–17000 g/mol) and a relatively low molecular-weight distribution (M_w/M_n = 1.6–1.8) are synthesized. Formation of the block copolymers is confirmed by ¹Н NMR spectroscopy, gel-permeation chromatography, and atomic force microscopy.     

The thermodynamic properties of the [(Me<Subscript>3</Subscript>Si)<Subscript>7</Subscript>C<Subscript>60</Subscript>]<Subscript>2</Subscript> fullerene complex

The temperature dependence of the heat capacity C _p ^o of the [(Me₃Si)₇C₆₀]₂ fullerene complex was measured for the first time using precision adiabatic vacuum calorimetry over the temperature range 6.7–340 K and high-accuracy differential scanning calorimetry at 320–635 K. For the most part, the error in the C _p ^o values was about ±0.5%. An irreversible endothermic effect caused by the splitting of the dimeric bond between fullerene fragments and the thermal decomposition of the complex was observed at 448–570 K. The thermodynamic characteristics of this transformation were calculated and analyzed. Multifractal analysis of the low-temperature (T < 50 K) heat capacity was performed, and conclusions were drawn concerning the character of the heterodynamicity of the structure. The experimental data obtained were used to calculate the standard thermodynamic functions C _p ^o (T), H ^o (T) ? H ^o (0), S ^o (T) ? S ^o (0), and G ^o (T) ? H ^o (0) over the temperature range from T → 0 to 445 K and estimate the standard entropy of formation of the compound from simple substances at 298.15 K. The standard thermodynamic properties of [(Me₃Si)₇C₆₀]₂ are compared with those of the (C₆₀)₂ dimer, the [(η⁶-Ph₂)₂Cr]⁺[C₆₀]^?? fulleride, and the initial C₆₀ fullerene.     

Perfect matchings in random pentagonal chains

Let G be a (molecule) graph. A perfect matching, or kekulé structure and dimer covering, in a graph G is a set of pairwise nonadjacent edges of G that spans the vertices of G. In this paper, we obtained the explicit expression for the expectation of the number of perfect matchings in random pentagonal chains. Our result shows that, for any polygonal chain \(Q_{n}\) with odd polygons, the number of perfect matchings can be determined by their concatenation LA-sequence.     

Distance based indices in nanotubical graphs: part 1

Nanotubical structures are obtained by wrapping a hexagonal grid, and then possibly closing the tube with caps. We show that the size of a cap of a closed (k, l)-nanotube is bounded by a function that depends only on k and l, and that those extra vertices of the caps do not influence the obtained asymptotical value of the distance based indices considered here. Consequently the asymptotic values are the same for open and closed nanostructures. We also show that the asymptotic for Wiener index, Schultz index (also known as degree distance), and Gutman index for (all) nanotubical graphs of type (k, l) on n vertices are \(\frac{n^3}{6(k+l)}+O(n^2)\), \(\frac{3n^3}{2(k+l)}+O(n^2)\), and \(\frac{n^3}{k+l}+O(n^2)\), respectively. In all cases, the leading term depends on the circumference of the nanotubical graph, but not on its specific type. Thus, we conclude that these distance based topological indices seem not to be the most suitable for distinguishing nanotubes with the same circumference and of different type as far as the leading term is concerned.     

Crystal structure of KAsUO<Subscript>6</Subscript>·<Subscript>3</Subscript>H<Subscript>2</Subscript>O at 100 and 293 K

The crystal structure of KAsUO₆·3H₂O was solved at 100 K and 293 K. KAsUO₆·3H₂O at T = 100 K: tetragonal, P4/ncc, a = 7.2037(6) Å, c = 17.811(2) Å; Z = 4, R1 = 0.0263, wR2 = 0.0546, for 618 independent reflections; at T = 293 K: tetragonal, P4/ncc, a = 7.1600(4) Å, c = 17.746(1) Å; Z = 4, R1 = 0.0263, wR2 = 0.0433 for 427 independent reflections. The results of X-ray analysis are compared with our previous data on heat capacity of this compound, and changes that take place in the structure at elevated temperature are considered.     

Crystal structures of [Zn(SALIMP)(CH<Subscript>3</Subscript>CO<Subscript>2</Subscript>)]<Subscript>2</Subscript> and [Cu(SALIMP)Cl] with 2-[[(2-pyridinylmethyl) imino]methyl]phenol (HSALIMP) as a ligand

Two complexes [Zn(SALIMP)(CH₃CO₂)]₂ (1) and [Cu(SALIMP)Cl] (2) are obtained by the reactions of zinc(II) and copper(II) salts with a tridentate Schiff base ligand 2-[[(2-pyridinylmethyl) imino]methyl]phenol (HSALIMP). Their structure is determined by single crystal X-ray diffraction. Data for complex 1: C₃₀H₂₈N₄O₆Zn₂, CCDC number: 668213, M _r = 671.3, monoclinic, C2/c, with a = 34.670(5) Å, b = 15.266(2) Å, c = 23.464(4) Å, β = 114.045(2)°, V = 11341(3) ų, Z = 16, F(000) = 5504, GOOF(F ²) = 0.894, the final R = 0.0520 and wR = 0.1272 for 10515 observed reflections with I > 2σ(I); complex 2: C₁₃H₁₂N₂OClCu, CCDC number: 668211, M _r = 311.24, triclinic, P-1, with a = 7.4050(8) Å, b = 10.2369(11) Å, c = 16.2873(17) Å, α = 87.728(2)°, β = 87.818(2)°, γ = 78.279(2)°, V = 1207.4(2) ų, Z = 4, F(000) = 632, GOOF(F ²) = 1.077, the final R = 0.0326 and wR = 0.0381 for 4209 observed reflections with I > 2σ(I).     

Synthesis of water soluble metalloporphyrin-cored amphiphilic star block copolymer photocatalysts for an environmental application

Two series of water-soluble metalloporphyrin-cored amphiphilic star block copolymers were synthesized by controlled radical polymerizations such as atom transfer radical polymerization (ATRP) and reversible addition fragmentation chain transfer (RAFT), which gave eight amphiphilic block copolymer arm chains consisting of poly(n-butyl acrylate-b-poly(ethylene glycol) methyl ether methacylate) (PnBA-b-PEGMEMA, M_n,GPC = 78,000, M_w/M_n = 1.2, 70 wt% of PPEGMEMA) and poly(styrene-b-2-dimethylamino ethyl acrylate) (PS-b-PDMAEA, M_n,GPC = 83,000, M_w/M_n = 1.2, 67 wt% of PDMAEA), yielding porphyrin(Pd)-(PnBA-b-PPEGMEMA)₈ and porphyrin(Pd)-(PS-b-PDMAEA)₈, respectively. Obtained metalloporphyrin polymer photocatalysts were homogeneously solubilized in water to apply to the removal of chlorophenols in water, and was distinguished from conventional water-insoluble small molecular metalloporphyrin photocatalysts. Notably, we found that the water-soluble star block copolymers with hydrophobic–hydrophilic core–shell structures more effectively decomposed the chlorophenol, 2,4,6-trichlorophenol (2,4,6-TCP), in water under visible light irradiation (k = 1.39 h^?1, t_1/2 = 0.5 h) in comparison to the corresponding water-soluble star homopolymer, because the hydrophobic core near the metalloporphyrin effectively captured and decomposed the hydrophobic chlorophenols in water.     

Ionic complexes simultaneously containing fullerene anions and coordination structures of metal phthalocyanines with I<Superscript>−</Superscript> and EtS<Superscript>−</Superscript> anions

The ionic complexes simultaneously containing negatively charged coordination structures of metal phthalocyanines and fullerene anions, viz., {Mn^IIPc(CH₃CH₂S^?) _x ·(I^?)_1?x }·(C₆₀ ^·?)· ·(PMDAE⁺)₂·C₆H₄Cl₂ (PMDAE is N,N,N′,N′,N′-pentamethyldiaminoethane, x = 0.87, 1) and {Zn^IIPc(CH₃CH₂S^?)_y·(I^?)_1?y }₂·(C₆₀ ^?)₂·(PMDAE⁺)₄·(C₆H₄Cl₂) (y = 0.5, 2) were synthesized. The both compounds were obtained as single crystals, which made it possible to study their crystal structures. In complex 1, the fullerene radical anions form honeycomb-like layers in which each fullerene has three neighbors with center-to-center interfullerene distances of 10.13–10.29 Å. Rather long distances between the C₆₀ ^·? radical anions results in the retention of monomeric C₆₀ ^·? in this complex down to the temperature of 110(2) K. In complex 2, fullerenes form dimers (C₆₀ ^?)₂ bonded by one C-C bond. The dimers are packed in corrugated honeycomb-like layers with interfullerene center-to-center distances of 9.90–10.11 Å. Manganese(II) and zinc(II) phthalocyanines coordinate iodide and ethanethiolate anions to the central metal atom to form unusual negatively charged coordination structures M^IIPc(An^?) (An^? is anion) packed in dimers {M^IIPc(An^?)}₂ with a short distance between the phthalocyanine planes (3.14 Å in 1 and 3.27 Å in 2). The pthalocyanine dimers also form layers with the PMDAE⁺ cations, and these layers alternate with the fullerene layers. The packing of spherical fullerenes with planar phthalocyanine molecules is attained by the insertion of fullerenes between the phenylene groups of phthalocyanines. The π-π-interactions of the porphyrin macrocycle with five- or six-membered fullerene rings are characteristic of the earlier studied ionic porphyrin and fullerene complexes. Such interactions are not observed for ionic complexes 1 and 2.     

Crystal Chemistry of Lithium Intermetallic Compounds: A Survey

The structural chemistry of lithium intermetallic compounds that are formed in Li–М binary systems where М = Ca, Sr, Ba, Mg, Zn, Cd, and Hg is surveyed. It is for the first time that the crystal structures of intermetallic compounds are classified in terms of polyhedral precursor metal clusters (in the program package ToposPro). The precursor metal clusters of crystal structures are identified using the algorithms of partitioning structural graphs into cluster structures and via the design of the basal 3D network of the structure in the form of a graph whose nodes correspond to the positions of the centers of precursor clusters. Tetrahedral precursor metal clusters M₄ are identified for the crystal structures LiZn₃-oC4, LiMg₃-hP2, LiCd₃-hP2, LiHg₃-hP8, (LiMg₃)(Li₂Mg₂)-tI16, Li₂Zn₂-cF16, Li₂Cd₂-cF16, Li₂Hg₂-cP2, Li₃Cd-cF4, and Li₃Hg-cF16; tetrahedral metal clusters M₄ are found for the framework structures with spacer atoms Sr(Li₂Sr₂)-tP20, Ca₂(Li₄)-cF24, and Ca₂(Li₄)-cP12; tetrahedral metal clusters M₄ and rings M₆, for framework structures Ba₃Li₂(Li₁₀)-hP30 and Ba₃Li₂(Li₄In₆)-hP30; icosahedral metal clusters M13 for the framework structure Li(Zn₁₃)-cF112; bilayer tetrahedral metal clusters 0@М₄@M₂₂ for the framework structure Li₂₃Sr₆-cF116; and deltahedra М₁₇ and deltahedra М₃₀, for framework structures Sr₄Li₁₄ [Sr(Sr₄Li₁₂)] [(Sr₂ (Sr₈Li₁₈)]-tI252 and Ba₄Li₁₄ [Ba(Ba₄Li₁₂)][(Ba₂ (Ba₈Li₁₈)]-tI252. The scenario of crystal structure self-assembly from precursor metal clusters S₃⁰ in intermetallic compounds is reconstituted as: primary chain S₃¹→ microlayer S₃²→ microframework S₃³.     

Paramagnetic centers related to Al,Sc, In,and Nb impurities in KTiOAsO<Subscript>4</Subscript>crystals

Electron paramagnetic resonance (EPR) is applied to study Al-, Sc-, In-, and Nb-doped KTiOAsO₄ (KTA) crystals. Paramagnetic hole centers O^? are observed after ionizing irradiation of KTA crystals. These centers are, as a rule, unstable at room temperature and are slowly annealed for about two weeks. Oxygen ions are bridging two cations in KTA. Near the impurity, two p-orbitals of oxygen atoms participate in covalent bonding with cations, whereas the third p-orbital remains free and under the radiation effect captures the hole thus forming the paramagnetic center of M ⁿ⁺-O^?-M^(n?1)+ (here M ⁿ⁺ is the lattice cation and M^(n?1)+ is the impurity cation of Al, In, Sc, or Nb). In the centers investigated the specific principal direction of the g-factor g ～ 2 is normal to the M ⁿ⁺-O^?-M^(n?1)+ plane, and the main value of g _max falls in this plane. The direction of the O^?-M^(n?1)+ bond is close to the selected direction of the hyperfine interaction with the impurity ion. The models of six hole centers and the found parameters of EPR spectra are discussed.     

Syntheses and characterization of two 1D Cu(II) coordination polymers with 2,2′:6′,2″-terpyridine ligand

The solution reactions of CuCl₂ with 2,2′:6′,2″-terpyridine (Terpy) and dicarboxylic acid (glutaric acid or suberic acid) afforded two 1D coordination polymers [Cu(Terpy)(C₅H₆O₄)] _n (I) and [Cu(Terpy)(C₈H₁₂O₄)] · 3H₂O (II) and their structures were characterized by IR, TG-DTA, and single-crystal X-ray diffraction. Crystallographic data for I: C₂₀H₁₆CuN₃O₄, M _r = 425.90, monoclinic, space group C2/c, a = 10.335(2), b = 21.193(4), c = 8.580(2) Å, β = 111.99(3)°, V = 1742.5(6) ų, Z = 4, ρ _c = 1.623 g/cm³, F(000) = 872, R = 0.0304 and wR = 0.0915; and those for II: C₂₃H₂₉CuN₃O₇, M _r = 523.03, triclinic, space group P \(\bar 1\), a = 8.362(2), b = 10.605(2), c = 14.617(3) Å, α = 73.26(3)°, β = 86.23(3)°, γ = 69.45(3)°, V = 1161.3(4) ų, Z = 2, ρ _c = 1.496 g/cm³, F(000) = 546, R = 0.0636 and wR = 0.1106. X-ray diffracion studies reveal that both complexes I and II feature 1D chain. The 1D polymer chains are connected by π-π-stacking interactions to generate 2D supramolecular layers.     

Crystal structure of Iron(III) hydrogen diphosphate FeHP<Subscript>2</Subscript>O<Subscript>7</Subscript>

The crystal structure of the β modification of iron(III) hydrogen diphosphate FeHP₂O₇ has been refined by the Rietveld method using powder X-ray diffraction data. The compound crystallizes in the monoclinic system, space group P2₁/n, Z = 4, a = 7.9756(1) Å, b = 12.8260(2) Å, c = 4.8664(6) Å, β = 98.6404(8)°, V = 492.16(1) ų. The structure was refined in the isotropic approximation (pseudo-Voigt function), R _p = 0.024, R _wp = 0.033, R _Bragg = 0.091, R _F = 0.067, and compared with the structures of other compounds M^IIIHP₂O₇ (M^III is a trivalent metal).     

Crystal structure of a cobalt oxide LuBaCo<Subscript>4</Subscript>O<Subscript>7</Subscript>

Crystals of LuBaCo₄O_7+δ (114Lu) have been obtained by spontaneous crystallization of slowly cooled nonstoichiometric melt of the system Lu-Ba-Co-O. The crystals have been characterized by EDX microprobe and synchrotron radiation powder diffraction. Structure refinement has been carried out (automated diffractometer Bruker X8 APEX with a CCD detector, MoK _α, graphite monochromator, ?_max = 32.54°). Parameters of the hexagonal unit cell: space group P6₃ mc, a = 6.2601(1) Å c = 10.2017(3) Å V = 346.23(1) Å₃, Z = 2, d _calc = 6.331 g/cm₃. Framework structure of the wurtzite type has been refined anisotropically to R-factor 0.0174. Charge balance in the compound and geometrical matching of structural fragments are discussed.     

Structure of the 1:1 complex of dimethyl sulfoxide with hydroquinone: Supramolecular architecture based on D-H⋯A hydrogen bonds (D = O,C; A = O, π)

The hydroquinone-dimethyl sulfoxide-toluene system was investigated by thermal and X-ray diffraction analyses. The crystal structure of the 1:1 complex of hydroquinone with dimethyl sulfoxide was determined. Crystal data: C₈H₁₂O₃S, M = 188.24, triclinic system, space group P1¯, unit cell parameters: a = 7.4202(2) Å, b = 8.4046(3) Å, c = 8.7340(3) Å; α = 100.830(1)°, β = 99.794(1)°, γ = 114.129(1)°; V = 469.35(4) ų, Z = 2, d _calc = 1.332 g/cm³, R1 = 0.028, T = 100 K. The molecules are linked in a supramolecular assembly via D-H...A hydrogen bonds (D = O, C; A = O, π).     

Influence of the Molecular Mass of Poly(<Emphasis Type="Italic">N</Emphasis>-vinylpyrrolidone) on Formation of Cu<Subscript>2</Subscript>O Nanoparticles During Reduction of Divalent Copper Ions with <Emphasis Type="Italic">tert</Emphasis>-Butylamine Borane in Polymer Solution

The process of reduction of divalent copper ions with tert-butylamine borane in dilute aqueous solutions of poly(N-vinylpyrrolidone) is investigated. The influence of polymer molecular mass on properties of the resultant Cu₂O sols is studied. It is shown that Cu₂O nanoparticles with an average diameter of 6–8 nm independent of polymer molecular mass and a relatively narrow size distribution of particles are formed in the systems under study. The contour length of macromolecules and the hydrodynamic diameter of a poly(N-vinylpyrrolidone) macromolecular coil are compared with the diameter of Cu₂O particles. Poly(N-vinylpyrrolidone) with M ≥ 1 × 10⁴ can be used to produce Cu₂O nanoparticles. Poly(N-vinylpyrrolidone) with M > 4 × 10⁴ should be used for the formation of long-living Cu₂O sols.     

Crystal structure of magnesio-ferri-hornblendite □Ca<Subscript>2</Subscript>(Mg<Subscript>4</Subscript>Fe<Superscript>3+</Superscript>)[(Si<Subscript>7</Subscript>Al)O<Subscript>22</Subscript>](OH)<Subscript>2</Subscript> as a potentially new mineral of the amphibole supergroup

A sample of magnesio-ferri-hornblendite, a potential new mineral of the amphibole supergroup, was studied by X-ray diffraction and IR spectroscopy. The crystal chemical formula is (Z = 2): ^AK_0.04^M(4) (Ca_1.92Na_0.08) ^C[^M(1)(Mg_1.78Fe_0.22⁴⁺) ^M(2)(Mg_1.62Fe_0.26³⁺Al_0.12) ^M(3)(Mg_0.64Fe_0.32²⁺Mn_0.04)] [^T(Si_7.44Al_0.56)O₂₂] ^W(OH)₂. The monoclinic unit cell parameters are a = 9.855(1) Å, b = 18.084(1) Å, c = 5.289(1) Å, β = 104.853(2)°; V = 911.1(2) ų; space group C2/m; Z = 2. The crystal structure was refined to R = 2.82% in the anisotropic approximation for atomic displacement parameters using 1166 reflections with I > 2σ(I). The magnesio-ferri-hornblendite structure is generally similar to the structures of other monoclinic calcium amphiboles, and its key distinctive features are the predominance of Мg among C²⁺ cations and Fe³⁺ among C³⁺ cations.     

On the origins of intersecting potential energy surfaces

Techniques developed for investigating nonadiabatic processes in molecular systems are adapted to study the structure and properties of holomorphic and meromorphic functions of a complex variable, \(f(z)=\mathfrak {R}(f)+i\,\mathfrak {I}(f)\). The connection is that \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) are correlated two-dimensional scalar functions, interrelated by the Cauchy–Riemann equations. Exploiting this fact, it is demonstrated that \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) of f can be envisaged in Euclidean \({\mathbb {R}}^{3}\) space as a two-state set of constrained, intersecting two-dimensional potential energy surfaces (PESs), called the graph of f. Importantly, the analytic and algebraic properties of f dictate the geometric structure evinced in the graph of f. This parallels multi-state sets of higher-dimensional, constrained, intersecting PESs linked with correlated electronic eigenstates of the parameterized molecular Hamiltonian operator. In view of this association, the language and mathematical infrastructure devised by chemists for discussing and analyzing intersections in higher-dimensional PESs are suitably modified for f. Notably, an algorithm capable of optimizing roots and poles of f through analysis of the real, two-dimensional \(\mathfrak {R}(f)\) and \(\mathfrak {I}(f)\) functions is derived, which is based on intersection-adapted coordinate and constrained Lagrangian methodologies. As constrained, intersecting PESs are indispensible for conceptualizing and characterizing the physics governing nonadiabatic phenomena, f represents a foundational bridge to these more abstract constructions.