Summary of Key Physico-chemical Properties of Phenol
Phenol, represented as PhOH, is the simplest aromatic alcohol and serves as the parent compound of a series of related substances that feature a hydroxyl group directly attached to an aromatic ring The presence of the hydroxyl group imparts acidity to phenol, while the benzene ring contributes to its basicity, making it the enol form of the carbonyl group.
Phenol is characterized by its low melting point and crystallizes into colorless prisms, emitting a slightly pungent odor In its molten state, it appears as a clear, colorless, and mobile liquid Its miscibility with water is limited below 68.4 °C, but it becomes completely miscible at higher temperatures The presence of water significantly lowers the melting and solidification points of phenol A mixture of phenol with 10% water is referred to as phenolum liquefactum, remaining liquid at room temperature Additionally, phenol is highly soluble in various organic solvents, including aromatic hydrocarbons, alcohols, and ethers, while being less soluble in aliphatic hydrocarbons It also forms azeotropic mixtures with water and other substances.
CHART 1 Chemical formulae of phenol: C6H5OH; early name: carbolic acid, hydroxybenzene;CAS registry number: 108-95-2
Other physical data of phenol follow below:
Molecular weight: 94.11 (molecular mass of C6H5OH is equal to 94.04186).
Weakly acidic: pK a (H2O)=9.94 (although it varies in different sources from 9.89 to 9.95).
Specific heats of combustion:C p =3.06 J mol −1 K −1 , C v =3.07 J mol −1 K −1
First ionization energy (IE a ): 8.47 eV (experimental), 8.49±0.02 eV (evaluated).
Proton affinity(PA): 820 kJ mol −1 10
Gas phase basicity: 786.3 kJ mol −1 10
Gas-phase heat of formation f H 298 :−96.2±8 kJ mol −1 (experimental);−93.3 kJ mol −1 (theoretical) 11
Experimental:−27.7 kJ mol −1 12 ,−27.6 kJ mol −1 13
Theoretical:−17.3,−20.2,−16.4 kJ mol −1 (AMBER parameter 14 ),−19.7,−23.8,
Experimental: 1465.7±10 kJ mol −1 17, 18 ; 1461.1±9 kJ mol −1 18, 19 ;
1471±13 kJ mol −1 20 Theoretical: 1456.4 kJ mol −1 20
Experimental: 362±8 kJ mol −1 21 ; 363.2±9.2 kJ mol −1 22 ; 353±4 kJ mol −1 23 ;
376±13 kJ mol −1 24 ; 369.5 kJ mol −1 25 ; 377±13 kJ mol −1 26 Theoretical: 377.7 kJ mol −1 20
The theoretical review of phenol highlights its electronic subsystem, which contains 50 electrons Notably, the ground state of phenol is identified as a singlet closed-shell state, referred to as S₀.
Phenol can be viewed as the enol form of cyclohexadienone, with its tautomeric equilibrium strongly favoring the enol side, unlike aliphatic ketones which lean towards the ketone form This stabilization in phenol is primarily due to the formation of an aromatic system, where high resonance stabilization arises from the contributions of ortho- and para-quinonoid resonance structures Additionally, the formation of the phenolate anion benefits from these quinonoid resonance structures, which help to stabilize the negative charge effectively.
Phenol is more acidic than aliphatic alcohols and can form salts with aqueous alkali hydroxide solutions, which can be liberated at room temperature even with carbon dioxide Near its boiling point, phenol can displace carboxylic acids from their salts, leading to the formation of phenolates The presence of ortho- and para-quinonoid resonance structures facilitates various electrophilic substitution reactions, including chlorination, sulphonation, nitration, nitrosation, and mercuration However, introducing multiple nitro groups into the benzene ring is challenging due to phenol's sensitivity to oxidation, although nitrosation at the para position can occur at low temperatures Additionally, phenol reacts readily with carbonyl compounds in the presence of acid or base catalysts, producing hydroxybenzyl alcohols from formaldehyde and bisphenol A from acetone.
Acid catalysts are employed to purify synthetic phenol by removing olefinic impurities and carbonyl compounds, such as mesityl oxide These impurities can be polymerized into higher molecular weight compounds using small amounts of sulfuric acid or acidic ion exchangers, facilitating their separation from phenol through distillation.
Phenol reacts with diazonium salts to form colored compounds, which can be utilized for photometric detection of phenol, such as with diazotized 4-nitroaniline Salicylic acid, or 2-hydroxybenzoic acid, can be synthesized through the Kolbe–Schmitt reaction using sodium phenolate and carbon dioxide, while potassium phenolate yields the para compound Additionally, phenol can undergo alkylation and acylation using aluminum chloride as a catalyst, and methyl groups can be introduced via the Mannich reaction However, the production of diaryl ethers requires extreme conditions.
Phenol easily reacts with oxidizing agents to create stable free radicals, which can either dimerize into diphenols or be further oxidized into dihydroxybenzenes and quinones Due to the stability of phenol radicals, phenol serves as an effective radical scavenger and oxidation inhibitor However, this property can have negative implications, as evidenced by its ability to inhibit the autoxidation of cumene even in small amounts.
The History of the Discovery of Phenol
Phenol, originally known as 'carbolic acid' or 'coal oil acid,' is a key component of coal tar It was first partially isolated from coal tar in 1834 by the chemist Runge.
Friedlieb Ferdinand Runge (born in Billw¨arder, near Hamburg, 8 February
Johann Runge, born in Oranienburg in 1795 and who passed away on March 25, 1867, began his career as a pharmacist before becoming an associate professor in Breslau, Germany, after a lengthy stay in Paris He later served in the Prussian Marine in Berlin and Oranienburg Runge made significant contributions to science and technology, publishing various papers and books Notably, he rediscovered aniline in coal-tar oil, naming it kyanol, and also discovered quinoline (leukol), pyrrole, rosolic acid, and three additional bases.
Pure phenol was first synthesized by Auguste Laurent in 1841 Born on September 14, 1808, in La Folie, Haute-Marne, Laurent was the son of a wine merchant and served as an assistant to Dumas at the Ecole Centrale and Brongniart at the Sevres porcelain factory From 1835 to 1836, he operated a private laboratory in a garret on Rue St Andre in Paris Laurent earned his doctorate in Paris in December 1837 and became a professor in Bordeaux in 1838.
1845 he worked in a laboratory at the Ecole Normale in Paris In his studies of the distil- late from coal-tar and chlorine, Laurent isolated dichlorophenol (acide chloroph´en`esique)
C 24 H 8 Cl 4 O 2 and trichlorophenol (acide chloroph´enisique) C 24 H 6 Cl 6 O 2 , which both sug- gested the existence of phenol (phenhydrate) 33 Laurent wrote: ‘I give the name ph`ene
In 1841, Laurent made a significant contribution to chemistry by isolating and crystallizing phenol, which he referred to as ‘hydrate de phényle’ He also provided a table detailing the general formulas for the derived radicals of phenol, although he mistakenly represented phenol with the formula C24H12 + H4O2, which is equivalent to C6H8O in modern notation.
The substance 'acide phénique' has a melting point between 34 and 35 °C and a boiling point between 187 and 188 °C, which align closely with current values In addition to measuring its physical properties, Laurent distributed crystals of the substance to individuals suffering from toothache to test its potential as a pain reliever However, the results regarding pain relief were ambiguous, and the substance was noted to be "very aggressive on the lips and the gums."
In his experiments, Laurent expanded on the substitution hypothesis proposed by his former supervisor, Dumas, by asserting that the substitution reaction does not alter the structural formula of the reactant and product In contrast, Dumas only suggested that the removal of one hydrogen atom is balanced by the addition of another group, allowing for the possibility of a complete molecular rearrangement.
The substitution hypothesis, particularly in Laurent's formulation, faced significant criticism from Berzelius He argued that replacing a hydrogen atom with another element, such as chlorine, in an organic molecule would be entirely unfeasible.
Chlorine's strong electronegative character has led to significant debates in chemistry, particularly between Berzelius and Dumas Berzelius criticized Laurent's ideas as contradictory to fundamental chemical principles, suggesting they negatively impacted scientific progress He preferred to reinterpret Laurent's findings by decomposing reaction products into smaller, more familiar molecules, showing a reluctance to embrace the complexities of organic chemistry and the existence of new molecules In response, Dumas clarified that he never claimed chlorine replaced hydrogen, emphasizing that the law of substitution is simply an empirical observation linking expelled hydrogen to retained chlorine Dumas also distanced himself from Laurent's exaggerated interpretations of his theory, questioning the reliability of Laurent's analyses.
In 1843, Charles Frederic Gerhardt first synthesized phenol by heating salicylic acid with lime, naming it 'phénol.' Since then, phenol has been the focus of extensive research, particularly from Victor Meyer, who investigated compounds like desoxybenzoin and benzyl cyanide, revealing their similar reactivities He published findings on the 'negative nature of the phenyl group,' highlighting its role in enhancing the reactivity of hydrogen atoms in methylene groups In 1867, Heinrich von Brunck defended his Ph.D thesis in Tübingen on 'Derivatives of Phenol,' concentrating on the isomers of nitrophenol.
The Raschig-Dow process for manufacturing phenol from cumene was first discovered by Wurtz and Kekule in 1867, with earlier synthesis documented by Hunt in 1849 Friedrich Raschig, a chemist at BASF known for his contributions to phenol synthesis and phenol-formaldehyde production, later founded his own company in Ludwigshafen.
In 1905, the BAAS subcommittee on dynamic isomerism was formed, chaired by Armstrong, with Lowry as secretary and Lapworth as a member In their 1909 report, Lowry highlighted that one type of isomerism involves the 'oscillatory transference' of hydrogen atoms between different elements, such as from carbon to oxygen in ethyl acetoacetate, from oxygen to nitrogen in isatin, or between oxygen atoms in para-nitrosophenol.
Usage and Production
The Equilibrium Structure of Phenol in the Ground Electronic State
Until the mid-1930s, studies on phenol using electron diffraction or microwave techniques had not been conducted, leaving its equilibrium configuration uncertain despite indirect evidence suggesting a planar ground electronic state The first X-ray structural data for various phenolic compounds emerged in 1938, indicating that the C−O bond measures approximately 1.36 Å, which is about 0.07 Å shorter than the C−O bond in aliphatic alcohols This shortening is attributed to the carbon atom's reduced effective radius due to a hybridization change from sp³ to sp², alongside some degree of electron delocalization across the C−O bond The increased double-bond character supports a completely planar equilibrium configuration for phenol in its ground electronic state.
The character of this molecule arises from quinonoid resonance structures alongside the predominant Kekulé-type structures, which positions the hydrogen atom within the molecular plane This configuration results in two equivalent arrangements for the hydrogen atom.
The positioning of the OH group relative to the C−O bond indicates the presence of an activation barrier, V τ, for the torsional motion of the OH group around the C−O bond, which is estimated to be 14 kJ mol −1, based on studies from the mid-thirties.
The molecular geometry of phenol has been elucidated through experimental techniques such as microwave spectroscopy and electron diffraction Studies from 1960 indicated that phenol derivatives maintain a planar configuration, exhibiting Cs symmetry Subsequent research in 1966 identified two potential r-structures by analyzing isotopic variations, while a comprehensive r-structure was established in 1979 Overall, the phenyl ring structure in phenol shows minimal deviation from that of an isolated phenyl ring, with nearly equal C−H distances, though the para-distance appears shorter Additionally, the CCC bond angles exhibit slight perturbations, with the C1C3C5 angle exceeding 120° and the C2C6C4 angle falling below it The angle between the C6O7 bond and the C1–C4 axis has been measured at 2.52°, with computational predictions suggesting a value of 2.58°.
Since the first quantum mechanical calculation of phenol in 1967 using the CNDO/2 method, various computational approaches, including HF, MP2, and DFT with split valence basis sets like 6-31G(d,p) and 6-31+G(d,p), have been employed to study phenol's geometry The results, detailed in Tables 1-3 and Figure 4, indicate that semi-empirical geometries closely align with experimental data Additionally, the theoretical inertia moments of the phenol molecule calculated at the B3LYP/6-31+G(d,p) level are 320.14639, 692.63671, and 1012.78307 a.u The rotational constants in Table 2 further demonstrate a strong agreement between theoretical predictions and experimental findings, highlighting the accuracy of these computational methods in characterizing phenol's properties.
Q xxxz = Q yyyz = Q zzzx = Q zzzy Q xxyz = Q yyxz = 0
The planar B3LYP/6-31+G(d,p) phenol molecule exhibits key properties in its ground electronic state, including the center of mass position, Mulliken charges, and the total dipole moment direction The accuracy of these properties is supported by the MP2 and B3LYP methods, which show mean absolute deviations of less than 0.2%, while the B3P86 method has a deviation of less than 0.6% In contrast, the HF and BLYP methods yield significantly larger deviations of approximately 1.3% and 1.5%, respectively, highlighting their known limitations; specifically, HF predicts bond distances that are too short, whereas BLYP predicts them as too long Notably, the C−O bond length is reported as 1.384 Å by BLYP/6-31G(d), which is shorter than the MP2/6-31G(d) value of 1.396 Å, indicating a discrepancy of 0.012 Å.
Molecular Bonding Patterns in the Phenol S o
In the phenol molecule, bonding can be understood through the two σ bonds formed by the oxygen atom, which are derived from trigonal hybrids One of these hybrids contains a coplanar sp² lone pair, while the pure p orbital interacts with the π electrons of the phenyl ring At a more advanced theoretical level, the electronic structure of the closed-shell ground-state phenol is characterized by 25 occupied molecular orbitals (MOs), as partially illustrated in Figure 5 These MOs are divided into two categories: seven core orbitals, which are 1s atomic orbitals from carbon and oxygen, and 18 valence orbitals that include six σ C−C bonds.
TABLE1.Phenolgeometry.Bondlengthsin ˚ A, bondanglesindeg E xperiment T h eory MW 107 MW 108 ED 109 MNDO a MINDO/3 a AM1 a PM3 a HF/ST O -3G 111 HF/4-31G 111 HF/6-31G 111 HF/6-31G(d) 111 HF/6- 31G(d,p) 112 HF/ DZP 113 Bond lengths C 1 − C 2 1.398 1.3912 1.3969 1.420 1 419 1.402 1.401 1.397 1.381 1 385 1.385 1.410 1.389 C 2 − C 3 1.3944 1.3969 1.405 1 406 1.394 1.390 1.386 1.385 1 389 1.387 1.392 C 3 − C 4 1.3954 1.3969 1.405 1 404 1.394 1.390 1.390 1.381 1 385 1.382 1.387 C 4 − C 5 1.3954 1.407 1 408 1.397 1.392 1.384 1.387 1 390 1.388 1.393 C 5 − C 6 1.3922 1.403 1 403 1.391 1.388 1.382 1.389 1 383 1.381 1.386 C 1 − C 6 1.3912 1.423 1 424 1.406 1.402 1.392 1.383 1 386 1.388 1.393 C 1 − O 7 1.364 1.3745 1.3975 1.359 1 326 1.377 1.369 1.395 1.374 1 377 1.352 1.382 1.354 C 2 − H 8 1.0856 1.081 1.090 1 105 1.099 1.096 1.082 1.073 1 074 1.077 C 3 − H 9 1.076 1.0835 1.091 1 106 1.100 1.095 1.083 1.072 1 073 1.075 1.093 C 4 − H 10 1.082 1.0802 1.90 1 104 1.099 1.096 1.082 1.071 1 072 1.074 1.092 C 5 − H 11 1.0836 1.091 1 107 1.100 1.096 1.083 1.072 1 072 1.075 C 6 − H 12 1.0813 1.090 1 104 1.099 1.096 1.082 1.069 1 070 1.074 O 7 − H 13 0.956 0.9574 0.953 0.948 0 951 0.968 0.949 0.989 0.950 0 949 0.947 0.977 0.944 Bond angles C 1 C 2 C 3 119.43 118.77 119.6 119 5 119 1 119 0 119 9 119 6 119 4 119 6 C 2 C 3 C 4 120.48 120.57 120.6 121 0 120 4 120 4 120 5 120 5 120 4 120 5 C 3 C 4 C 5 119.74 119.75 119.8 119 1 120 0 120 1 119 4 119 4 119 4 119 2 C 4 C 5 C 6 120.79 120.7 121 4 120 6 120 6 120 8 120 7 120 6 120 7 C 1 C 6 C 5 119.22 119.4 119 0 118 9 118 9 119 6 119 6 119 4 119 5 C 1 C 2 H 8 120.01 121.2 121 3 120 4 120 9 120 4 120 2 120 3 120 0 C 2 C 3 H 9 119.48 119.5 119 1 119 5 119 6 120 3 119 4 119 5 119 4 C 3 C 4 H 10 120.25 120.1 120 5 120 1 120 0 120 4 120 3 120 3 120 4 C 6 C 5 H 11 119.43 119.5 118 9 119 5 119 6 C 1 C 6 H 12 119.23 120.8 121 7 119 5 120 4 C 1 O 7 H 13 109 0 108.77 106.4 112.8 114 0 107 9 107 9 104 9 114 8 114 7 110 7 108.1 1 10 9
TABLE1.(continued) Th eo ry CAS(8,7) cc- pVDZ 114
B3L Y P/ cc-pVDZ 114 MP2/ DZP 113 MP2/ 6-31G (d,p) 111
TABLE2.Rotationalconstants(inMHz)ofphenolinitselectronicgroundstate.Thevaluesinparenthesesarethedeviationfromtheexperimental valuesinpercent HF/6- 31G(d,p) 124 HF/6- 311 ++ G (d,p) 124
CAS(8,7)/ cc-pVDZ 126 MP2/6- 31G(d,p) 124 BL YP/6- 31G(d,p) 124 B3L Y P/6- 31G(d,p) 124 B3L Y P/6- 31 + G(d,p) a B3L Y P/6- 311 ++ G(d,p) a B3L Y P/6- 311 ++ G (2df,2p) a
B3P86/6- 31G(d,p) 124 MW E xpt 108 UV E xpt 118 UV E xpt 127 A 5750.0 5752.6 5659.3(0.16) 5650.6 5563.7 5650.4 5637.3 5667.2 5695.6 5679.9 5650.5154 5726 63 5650.515 B 2659.1 2660.0 2623.3(0.16) 2614.6 2573.7 2614.1 2607.3 2618.0 2629.6 2630.0 2619.2360 2660 0 2619.236 C 1818.3 1818.9 1792.4(0.14) 1787.5 1759.7 1787.3 1782.8 1790.8 1799.0 1797.3 1789.8520 1820 12 1782.855 a Pres ent w ork.
TABLE 3 Dipole moment of phenol Experimental data are partly reproduced from
Experiment Theory à(D) Phase of solvent b T( ◦ C) à(D) Method
1.53 128 B a See also Figure 4. b B = benzene, D = dioxane, c-Hx = cyclohexane, Hp = n-heptane, Tol = toluene, ClB = chlorobenzene;n.s = not specified.
The molecular orbital patterns of the electronic ground state of the phenol molecule exhibit Cs symmetry, which is reflected in its molecular orbitals characterized by the irreducible representations of this symmetry group The symbol ε represents the corresponding orbital energy measured in electron volts (eV).
24a ′′ e = −7.10 eV 25a′′-HOMO e = −6.33 eV 26a′′-LUMO e = −0.51 eV
The molecular structure includes five σ C−H bonds and a C−O σ bond, alongside oxygen's σ-type and p-type lone pairs Additionally, there are C−C π-bonds represented by three a orbitals, specifically 23a, 24a, and the HOMO 25a Figure 5 also illustrates three unoccupied π molecular orbitals, identified as LUMO 26a, 27a, and 28a, with energy values of −0.12 eV and −0.09 eV respectively.
Table 4 showcases the natural atomic charges and total populations of core, valence, and Rydberg electrons for each atom Notably, the hydroxyl hydrogen atom H13 exhibits a higher positive charge compared to other atoms, attributed to its proximity to the electronegative oxygen atom In contrast, the adjacent hydrogen atom H8 displays the lowest positive charge, which results from electron donation from the ring to the C−H antibonding orbital, effectively reducing electrostatic repulsion between neighboring C−H bonds.
The Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) are crucial in understanding molecular behavior Figure 5 illustrates that the HOMO shape results from the out-of-phase overlap of pz atomic orbitals (AOs), highlighting its localized nature.
TABLE 4 Natural atomic orbital (NAO) occupancies, natural population of the MOs, summary of natural population analysis and Mulliken atomic charges of the electronic ground state of phenol
N Atom N lm Type(AO) Occupancy Energy (eV)
N Atom N lm Type(AO) Occupancy Energy (eV)
Natural population of the MOs
Summary of natural population analysis
Atom N Charge Core Valence Rydberg Total
The analysis of phenol reveals significant details about its electronic structure, particularly concerning the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) The HOMO, influenced by carbon atoms C1, C2, and C6, along with C4 and an oxygen atom, exhibits two nodal surfaces that are perpendicular to the phenolic ring, with an energy of ε HOMO = −6.33 eV In contrast, the LUMO, characterized by the out-of-phase overlap of p z atomic orbitals on C2, C3, C5, and C6, has an energy of ε LUMO = −0.51 eV According to Koopmans’ theorem, the ionization potential can be approximated by the negative HOMO energy, suggesting that ε HOMO represents the energy needed to remove a π electron from phenol, forming the phenol radical cation The experimental adiabatic first ionization energy (IE a) of phenol is measured at 8.49±0.2 eV, which is notably lower by approximately 71 kJ mol −1 compared to that of benzene This indicates that Koopmans’ theorem may not be a reliable predictor for phenol's vertical ionization energy.
To theoretically calculate the ionization energy of phenol, it is essential to apply the same methodology for both the parent compound and its cation Table 5 presents a summary of the optimized geometries and energies, including zero-point vibrational energy (ZPVE), for phenol and its radical cation, determined using the B3LYP method with the 6-31G(d,p) basis set.
The 311++G(d,p) basis sets reveal a significant alteration in the geometry of the phenol radical cation when compared to the original phenol molecule, particularly around the carbonyl group Notably, the variation between the vertical ionization energy (IEvert) and the adiabatic ionization energy (IEad) is also highlighted.
TABLE 5 The B3LYP data of phenol and phenol radical cation a,b
Geometry Phenol Phenol radical cation
7.03 HF/DZP 113 8.70 MP2/DZP 113 8.15 B3LYP/DZP 113
Recent theoretical studies have focused on the phenol radical cation, revealing various properties as detailed in Reference 139 The bond lengths, measured in angstroms, bond angles in degrees, energies in hartree, zero-point vibrational energy (ZPVE) in kJ mol−1, and ionization energy in electron volts (eV) are crucial for understanding its structure The atomic numbering is provided in Chart 1, with deviations in bond lengths from those of phenol noted in parentheses Additionally, the energy of the phenol radical cation is evaluated based on the geometry of the parent phenol, indicating minimal changes Further exploration of the potential energy surface of the ionized phenol will be addressed in the following section.
Atom-in-Molecule Analysis
In this subsection, we briefly review the use of the functionL(r) of the electronic ground-state phenol which is defined as minus the Laplacian of its electron density,
The Bader's Atoms in Molecules (AIM) approach provides a comprehensive understanding of the topology of one-electron density ρ(r), which is essential for defining 'atoms' within a molecule or molecular aggregate This is achieved through the gradient vector field, allowing for a nearly accurate mapping of the electron pairs in the VSEPR model Additionally, the electronic localization function plays a crucial role in this analysis, further enhancing our grasp of molecular structure.
A vector field represented by ∇ r ρ(r) consists of gradient paths that can be visualized as curves in three-dimensional space, tracing the direction of the steepest ascent in ρ(r) The concept of a gradient path is straightforward: it begins and concludes at specific points within the field.
∇ r ρ(r)vanishes These points are calledcritical points(CPs) The CPs ofρ(r)are special and useful points of the corresponding molecule.
The classification of the critical points is the following 142 There are three types of CPs:maximum, minimum or saddle point In 3D, one has two different types of saddle points.
Critical points (CPs) of the 3D function ρ(r) are classified based on the eigenvalues λi (i=1,2,3) of the Hessian matrix ∇²ρ(r), evaluated at each CP Each CP is represented by an (r,s) pattern, where r indicates the rank (the number of non-zero eigenvalues) and s represents the signature (the sum of the signs of the eigenvalues) For instance, a saddle point with two negative eigenvalues and one positive eigenvalue has a rank of r=3 and a signature of s=−1, designating it as a (3,−1) CP This type of CP, known as a bond critical point, signifies a bond between two nuclei in a molecule Bond critical points are connected to adjacent nuclei through an atomic interaction line, which consists of gradient paths originating at the bond CP and leading to each nucleus Collectively, these atomic interaction lines form the molecular graph of the molecule.
The AIM analysis of electron density and its Laplacian was conducted at the B3LYP/cc-pVDZ level using the MORPHY suite of codes The AIM charges obtained are presented in Table 6 Figures 6 and 7 illustrate the molecular graph L(r), showcasing various views of the one-electron density in the electronic ground state of phenol, highlighting regions of local charge concentration that correspond to maxima in the density.
The valence shell charge concentration (VSCC) graph of phenol reveals critical points (CPs) that indicate regions of local charge depletion As illustrated in Figure 6, the graph comprises a total of 87 CPs, including 27 (3,−3) CPs and 41 (3,−1) CPs, highlighting the geometric positions of these critical points within the molecular structure.
The (3,−3) critical points (CPs) in L(r) can be categorized into three distinct subsets: the non-bonding maxima associated with oxygen, the bonding maxima formed between pairs of carbon, oxygen, and hydrogen atoms, and the nuclear maxima that align closely with hydrogen nuclei The (3,−1) CPs serve a role similar to bond critical points, acting as connections between maxima By tracing the gradient paths in L(r) from the (3,−1) CPs, we typically expect these paths to link maxima, which holds true for most (3,−1) CPs in phenol However, there are instances where two (3,−1) CPs are connected near the oxygen atom, a phenomenon usually seen in 'conflict' structures This unusual connectivity indicates that a planar graph cannot be constructed for the Valence Shell Charge Concentration (VSCC).
The valence shell charge depletion (VSCD) graph of phenol, illustrated in Figure 7, reveals the geometric positions of various critical points (CPs) It features a total of 55 (3,−1) CPs, 80 (3,+1) CPs, and 22 (3,+3) CPs, showcasing a significantly more intricate structure compared to the valence shell charge concentration (VSCC) graph.
TABLE 6 AIM charges of the ground-state phenol
The VSCC graph for phenol illustrates the positioning of the oxygen atom, highlighted in red, alongside critical charge density critical points (CPs) Green spheres represent the (3,−3) maxima in the phenolic L(r), while violet spheres denote the (3,−1) CPs Additionally, yellow spheres indicate the (3,+1) CPs, and the graph includes lines that depict domain interactions.
In the study of phenol, the connection between two (3,−3) critical points (CPs) through a (3,−1) CP highlights the entire molecular structure While the separation of the VSCC and VSCD graphs is a conceptual tool, it significantly aids in visualizing their importance The gradient paths of the VSCC graph illustrate the connectivity of charge concentration maxima, known as attractors, while the gradient paths of the VSCD graph reveal the extensions of the basins surrounding these attractors Key properties of the atoms in phenol are summarized in Table 7.
The VSCD graph for phenol illustrates the molecular structure, with the oxygen atom highlighted in red and hydrogen in white The brown spheres represent the (3,+3) critical points, indicating minima within the phenolic landscape, while the purple spheres denote the (3,−1) critical points Additionally, the yellow spheres signify the (3,+1) critical points, providing a comprehensive view of the molecular interactions in phenol.
The (3,+1) CPs link the (3,+3) CPs via a pair of gradient paths shown in white, each of which is repelled by a (3,+3) CP
Vibrational Modes
The phenol molecule consists of 13 atoms and features 33 normal vibrational modes Its overtone and combination bands are active in the infrared spectrum Accurately assigning the fundamental vibrational modes of phenol in its electronic ground state is essential for understanding its molecular characteristics.
TABLE 7 The AIM properties of the ground-state phenol
The total dipole components of phenol, measured at −0.0001, 0.3853, and 1.0577, have a rich research history beginning in 1941 with the assignment of observed Raman bands above 600 cm −1 The initial infrared spectra analysis of phenol – OD occurred between 1954 and 1955, leading to fundamental vibration assignments based on earlier studies The lowest vibrational mode, known as mode 10b, was assigned to 242 cm −1 in 1960 and later to 241 cm −1, with a subsequent observation of 235 cm −1 in 1981 Recent determinations of mode 10b frequencies in phenol and phenol-d1 were found to be 225.2 and 211.5 cm −1, respectively, suggesting possible inaccuracies in previous assignments Over the past two decades, this mode has received limited attention, as predicted values from various ab initio methods tend to be lower than experimental results Various spectroscopic techniques, including UV-VIS, IR, and microwave spectroscopy, have been employed to examine the vibrational modes of ground-state phenol, which are summarized in Table 8 using nomenclatures from Wilson and Varsanyi Recently, the vibrational modes of phenol have served as benchmarks for testing ab initio and density functional methods, with Hartree-Fock calculations first conducted using the 6-31G(d,p) basis set, followed by an MP2 study with the same set.
The study of the phenol spectrum and the comprehensive assignment of its vibrational modes was conducted using Hartree-Fock (HF), MP2, and the density functional theory BLYP, along with the 6-31G(d,p) basis set (refer to Table 9).
Figure 8 illustrates the normal displacements, while Table 10 presents the associated vibrational assignments, highlighting the stretching ν OH mode at the end of both This localized mode is extensively researched due to its relevance in understanding the hydrogen-bonding capabilities of phenol Additionally, the second overtone of this mode in phenol and its halogen derivatives has been the subject of experimental studies.
The OH group of phenol participates in two additional modes, in-plane and out-of- plane bending vibrations The latter is also called the torsional modeτ OH observed near
In the infrared (IR) spectra of phenol vapor and dilute phenol solutions in n-hexane, a significant feature is observed around 300 cm−1, indicating the presence of hydrogen-bonded associations Additionally, both liquid and solid phenol-OD show a range of broad bands within the 600 – 740 cm−1 region, further highlighting the molecular interactions present in these states.
In the IR spectrum of phenol vapor, the first overtone of the torsional mode of the OH group was identified at 583 cm⁻¹ This finding enables the modeling of the torsional motion of the OH group in phenol, suggesting that it can be effectively described by this specific overtone assignment.
TABLE 8 Experimental (infrared and Raman) and theoretical vibrational spectra of phenol
The study reports key spectral data including ν(OH) values of 3087, 3091, and 3656 cm⁻¹, derived from the first and third overtone and the combination band with mode 1a Additionally, the first overtone of normal modes provides further insights into the calculated values The research also introduces a torsional potential model characterized by Vτ (1−cos 2θ)/2, where θ represents the torsional angle and Vτ denotes the barrier height Notably, the reduced moment of inertia is established at 1.19×10⁻⁴⁰ g cm², contributing to the understanding of molecular dynamics.
Theβ COH is the in-plane bending of the OH group placed at around 1175 – 1207 cm −1
In the IR spectrum of phenol vapor, a notable absorption band is observed at 1176.5 cm−1, which shifts to 910 cm−1 in dilute solutions following deuteriation, resulting in a broad absorption between 930 and 980 cm−1 in the crystal spectrum Initial HF/6-31G(d,p) calculations estimated this band at 1197.3 cm−1, with a scaled value of 1081 cm−1 Additionally, twenty-four vibrational modes of phenol are primarily associated with the phenyl ring modes, showing minimal sensitivity to substituent variations Conversely, six modes exhibiting significant motion of the phenyl and CO groups are notably affected by isotopic substitution of OH with OD.
1260 (1253), 814 (808), 527 (523), 503 (503), 398 (380) and 242 (241) cm −1 for phenol and phenol – OD (in parentheses), respectively.
Recent studies on the near-infrared (near-IR) spectra of phenol have primarily examined how solvent effects and hydrogen-bonding influence the frequency of the first overtone of the ν OH stretching vibration Research has documented the ν OH vibration frequencies for vibrational quantum numbers from ν=0 to ν=5 using photoacoustic spectroscopy Additionally, the near-IR spectrum of phenol in solution has been explored between 4000 and 7000 cm −1 through conventional FT-IR spectroscopy Non-resonant two-photon ionization spectroscopy has also identified vibrational transitions in this range, with many assigned to combinations involving the ν OH vibration and other fundamental modes of phenol A key focus in this research is to understand the cluster of peaks around 6000 cm −1, attributed to the first overtone of the ν CH vibrations of phenol – OH, as their fundamental vibrations occur at 3000 cm −1 The ν CH absorptions of phenol – OH and phenol – OD, along with their first and second overtones, are analyzed using a deconvolution procedure in the near-IR spectra.
TABLE 9 Theoretical assignments of the vibrational modes of phenol 112 Potential energy distri- bution (PED) elements are given in parentheses, frequencies in cm −1 , IR intensities in km mol −1 a
At a concentration of 0.1 M, phenol may form dimers and higher-order associates in solution, as indicated by the weak band observed in the fundamental region.
In phenol, the –OH group exhibits a strong absorption band at 3485 cm −1, while the –OD group shows a peak at 2584 cm −1, indicating the presence of dimeric forms Additionally, weaker and broader bands near 3300 and 2500 cm −1 are attributed to higher associates of phenol A very weak absorption band at 6714 cm −1 in the near-IR spectrum also confirms the existence of the dimer.
Three Interesting Structures Related to Phenol
STRUCTURES AND PROPERTIES OF SUBSTITUTED PHENOLS
Since the discovery of phenol 160 years ago, extensive research on halophenols has been conducted, primarily due to their importance in understanding hydrogen bonding These compounds exhibit a wide range of hydrogen bonding capabilities, with pK values varying from 10.2 to 0.4.
Intramolecular Hydrogen Bond in ortho-Halogenophenols
In 1936, Linus Pauling proposed the existence of two distinct rotational isomers, or rotamers, of ortho-chlorophenol to account for the observed splitting of the first overtone of its ν OH vibrational mode in CCl4 solution While phenol exhibits a sharp overtone peak at 7050 cm−1, ortho-chlorophenol displays a doublet at 7050 and 6910 cm−1, leading to a band splitting of ν OH (1) at 0 cm−1, with the higher wavenumber aligning with phenol Nearly twenty years later, further investigation revealed additional splitting of the fundamental ν OH mode.
In CCl4 solvent, a significant observation was made regarding the energy difference between the cis and trans conformers of o-ClC6H4OH, with the cis conformer being energetically favored due to the presence of an intramolecular hydrogen bond (O−H Cl), resulting in a gain of energy of 12.5 kJ mol−1 Pauling's estimation of the free energy difference based on peak area ratios was 5.8 kJ mol−1, with more precise values reported at 6.1 kJ mol−1 and 7.5 kJ mol−1 Our calculations align closely with the free energy difference of 14.2 – 16.3 kJ mol−1 in vapor, bounded by 16.3±3.0 kJ mol−1 and 14.3±0.6 kJ mol−1 Additionally, the trans conformer exhibits greater polarity (3.0 D vs 1.04 D), with the key distinction between the conformers being the intramolecular hydrogen bond, suggesting that the cis–trans energy difference can be interpreted as the energy of formation, albeit relatively weak for cis o-ClC6H4OH.
The analysis of Table 11, which details the harmonic vibrational modes of both conformers and their potential energy distribution, shows that the trans ν OH is calculated at 3835.4 cm −1, closely matching the ν OH of phenol in Table 10 In contrast, the cis conformer exhibits a red shift, consistent with hydrogen bonding theory, by cis–trans ν OH Cl i cm −1 This calculated red shift aligns well with experimental values ranging from 58 to 63 cm −1, depending on the solvent, although our findings indicate a smaller red shift of 91 cm −1 compared to Wulf and colleagues' observations for ν OH (1), potentially due to anharmonic effects Additionally, the relatively low value of the hydrogen bridge stretching vibration ν σ (mode 2 in Table 11) further suggests a weak intramolecular hydrogen bond in cis o-ClC 6 H 4 OH when compared to its trans counterpart.
In the two decades following Pauling's work, significant criticism emerged regarding earlier experimental results, with many suggesting that the higher frequency band was likely due to phenol impurities rather than the presence of trans isomers New experiments revealed that the absorption ratio was much smaller, approximately 1/56 (around 17.9×10 −3), yet still about three times larger than previously thought.
∆ Ecis-trans Cl = 12.5 kJ mol − 1 Cl cis ortho-FC 6 H 4 OH cis ortho-ClC 6 H 4 OH cis ortho-BrC 6 H 4 OH
The potential energy surface of o-chlorophenol, illustrating the cis-trans conversion, is depicted, with atom numbering following Chart 1 The five-member sub-ring sections of the cis ortho-halogenophenols featuring intramolecular hydrogen bonds are also presented, with bond lengths in angstroms and bond angles in degrees Despite advancements in the Pauling model and its experimental validation, the unresolved issues surrounding the cis-trans doublet paradigm persist, creating a significant disconnect between experiments from the late 1950s and contemporary high-level theoretical studies, particularly concerning o-fluorophenol.
In 1958, experimental verification indicated that the cis–trans doublet for o-FC6H4OH could not be detected, as the trans ν OH band was too weak for infrared (IR) experiments, with the cis–trans ν OH splitting estimated at less than 20 cm −1 Our calculations predict a cis–trans energy difference of 4 kJ mol −1, showing that the intramolecular O−H F hydrogen bond in o-FC6H4OH is 1.09 kJ mol −1 weaker than that in o-ClC6H4OH Additionally, the theoretical splitting for cis–trans ν OH is greater than 20 cm −1, aligning with IR experimental predictions Notably, the dipole moment of trans o-FC6H4OH (2.95 D) is nearly three times greater than that of its cis form (1.0 D).
The table presents the harmonic vibrational frequencies, infrared (IR) intensities, and assignments for cis and trans ortho-chlorophenols Key frequencies include 1155.60 cm⁻¹ and 1152.61 cm⁻¹, associated with various vibrational modes such as τ and γCCl Significant IR peaks at 2249.33 cm⁻¹ and 2239.72 cm⁻¹ highlight βCCl and βCO contributions Higher frequencies, such as 3262.81 cm⁻¹ and 3260.31 cm⁻¹, indicate complex vibrational interactions The data also reveals notable assignments at 4375.83 cm⁻¹ and 4317.79 cm⁻¹, emphasizing τOH and νCCl modes Frequencies exceeding 10,000 cm⁻¹, including 10672.60 cm⁻¹, demonstrate the intricate vibrational landscape of these compounds, with multiple contributions from different vibrational modes The detailed assignments and percentages of potential energy distribution (PED) provide a comprehensive understanding of the molecular vibrations present in ortho-chlorophenols.
The table presents a comprehensive overview of various infrared (IR) spectral frequencies and their corresponding assignments, detailing the intensity ratios and percentages for different molecular vibrations Key findings include significant frequencies such as 191.14 cm⁻¹ for νC3C4 and βC5H, and 201.18 cm⁻¹ for βC5H and βC4H The data also highlights notable vibrations like νCO at 221.27 cm⁻¹ and νC5C6 at 231.32 cm⁻¹, emphasizing the diversity of molecular interactions Additionally, frequencies around 293.17 cm⁻¹ reveal substantial νC6H and νC5H contributions, while 333.76 cm⁻¹ indicates a strong νOH absorption This detailed spectral analysis is crucial for understanding molecular behavior in various experimental conditions, including gas-phase and solution environments.
The article presents a detailed table of harmonic vibrational frequencies, infrared (IR) intensities, and assignments for both cis and trans ortho-fluorophenols Key vibrational modes include τOH, τrg, γCF, and βCO, with specific frequencies ranging from 1190.40 cm⁻¹ to 17105.29 cm⁻¹ Notably, the strongest IR intensity is associated with the τOH mode at 4396.31 cm⁻¹ for cis ortho-fluorophenol, while various combinations of vibrational modes, such as γC6H and νC1C2, illustrate the complex interactions within the molecular structure The data highlights the differences in vibrational characteristics between the cis and trans isomers, providing insights into their chemical behavior and potential applications in spectroscopy.
The data presented in Table 12 illustrates various assignments and their corresponding percentages of peak energy distribution (PED) across different molecular vibrations Notable assignments include β 1rg, βCOH, and multiple ν (stretching vibrations), with specific percentages indicating their significance in the molecular structure For example, assignments such as βC5H(34) and νC4C5(12) highlight the diversity of vibrational modes across the compounds analyzed The table also emphasizes the prevalence of νOH vibrations, indicating their critical role in the molecular interactions Overall, this comprehensive analysis provides valuable insights into the vibrational characteristics of the studied compounds, essential for understanding their chemical behavior.
The transition state between the cis and trans isomers of o-FC6H4OH has a slope of approximately 347 icm−1, similar to that of Cl, but with a barrier of 3.3 kJ mol−1, which is 2.2 kJ mol−1 lower than that of Cl Given that the energy barrier for cis-trans isomerization is lower for o-FC6H4OH than for o-ClC6H4, we would anticipate that the equilibrium constant kF cis trans is greater than kcis Cl trans, which is supported by the finding that kF cis trans is approximately 1×10−3 However, despite these findings, no infrared (IR) experiments have successfully demonstrated a cis-trans transition in o-FC6H4OH, raising the question of why this is the case.
The difference between older infrared (IR) experiments and contemporary high-level theories is particularly pronounced in the case of theo-Br-substituted phenols, as illustrated by their harmonic vibrational modes listed in Table 13 This allows for the straightforward calculation of the cis–trans ν Br OH values, which align well with experimental measurements that range from 74 to 93 cm −1, as shown in Tables 1 and 2.
Recent studies indicate that the strength of intramolecular hydrogen bonds in ortho-substituted phenols varies unexpectedly, with bromine (Br) exhibiting a slightly stronger bond than chlorine (Cl), contradicting established norms of hydrogen bond acceptor strength Experimental data reveals a free energy difference in vapor of 13.1±14.6 kJ mol−1, while the calculated cis–trans energy difference for ortho Br is 9.9 kJ mol−1 This leads to an equilibrium constant for cis Br to trans being significantly lower than that for cis Cl to trans, despite experimental evidence suggesting the opposite trend This phenomenon has been termed an 'anomalous' order in hydrogen bond strength, as summarized by Sandorfy and colleagues in 1963, who noted the lack of explanation for these findings Future quantum chemistry advancements may provide deeper insights into this intriguing behavior.
Equation 5 illustrates the energy differences (in kJ mol −1) associated with the formation of intramolecular hydrogen bonds between the left-hand and right-hand complexes This energy comparison is further supported by the red shift order of the cis–trans ν OH X values (in cm −1) presented in equation 6.
A comparison of equations 5 and 6 reveals that the cis–trans ν OH X is not proportional to the cis–trans E ortho X, indicating that the order reflects the van der Waals radii of halogen atoms—Br (1.85 ˚A) > Cl (1.75 ˚A) > F (1.47 ˚A)—rather than their electronegativity trend in Pauling units This suggests that in incis ortho-XC6H4OH, the strength of the O−Hã ã ãX intramolecular hydrogen bond decreases in the order of Br ≈ Cl > F, which contradicts the conventional understanding of intermolecular hydrogen bonds This discrepancy has been the subject of extensive research, and an explanation can be provided based on geometrical criteria, where the strength of the hydrogen bond is related to the elongation of the O−H bond length and the O−Hã ã ãX bond angle Notably, the strength of the intramolecular hydrogen bond in incis ortho-X-substituted phenols aligns with the order established in equations 5 and 6.
meta- and para -Halogenophenols
Introduction to the ELF
Nearly a decade ago, Becke and Edgecombe in their seminal paper 255 introduced the electron localization function (ELF) η( r ) of an arbitrary N-electron system
The equation for the kinetic energy density in a system, denoted as η(r), is expressed as η(r) = (1 + [(t - tW)/tTF]^2)^(-1) Here, t represents the kinetic energy density calculated as t = (1/2) ∑(Ni=1) |∇ψi|^2, using the Hartree-Fock or Kohn-Sham methods, where ψi (i = 1, , N) are the molecular orbitals The Weizsäcker kinetic energy density, tW[ρ(r)], is defined as (∇ρ)^2 / (8ρ), with the one-electron density given by ρ(r) = ∑(Ni=1) |ψi(r)|^2 Additionally, the Thomas-Fermi kinetic energy density, tTF[ρ(r)], is proportional to αTF[ρ(r)]^(5/3), where αTF = 3(6π^2)^(2/3) / 5, derived from the uniform electron gas approximation.
The ELF η(r) is characterized by a simple normalized Lorentzian-type form, with its values ranging from 0 to 1 An upper limit of η(r) = 1 indicates a fully paired electron system, aligning with the Weizsäcker kinetic energy density derived from the Pauli principle Conversely, a value of η(r) less than 1 suggests the presence of unpaired electrons, with η(r) = 1/2 corresponding to the full width at half maximum (FWHM) This condition occurs when t = t_W[ρ(r)] ± t_TF[ρ(r)], with the lower sign applicable when t_W[ρ(r)] is less than t_TF[ρ(r)].
Topology of the ELF
The topological analysis of the electron localization function (ELF) aims to create a robust mathematical model that reconciles the Lewis and VSEPR theories with quantum mechanics, addressing their inherent contradictions By establishing a mathematical bridge between chemical intuition and quantum mechanics, ELF analysis partitions three-dimensional space into regions that correlate with chemical properties Utilizing the framework of dynamical systems, particularly gradient dynamical systems, this approach employs a scalar potential function to facilitate the partitioning of molecular space, thereby enhancing our understanding of bonding within molecules.
The AIM 141 method utilizes the gradient dynamical field of charge density ρ(r) to identify atomic basins To demonstrate the presence of electronic domains, it is necessary to select a local function associated with pair-electron density However, due to their dependence on two spatial variables, pair-electron functions cannot be directly employed as potential functions.
The Electron Localization Function (ELF) is a local measure that quantifies the effectiveness of Pauli repulsion at specific points within molecular space Initially derived from the Laplacian of conditional probability, the ELF has since been reinterpreted to reflect local excess kinetic energy density resulting from the Pauli exclusion principle This reinterpretation enhances the physical understanding of the ELF and enables its application to any wave function, including the exact wave function Consequently, the ELF serves as a robust framework for analyzing molecular and crystal bonding.
In 1994, the concept of utilizing the gradient field of Electron Localization Function (ELF) for topological analysis of molecular space was introduced, aligning with the principles of Atoms in Molecules (AIM) theory ELF attractors define basins that can be classified as core basins, which encompass nuclei, or valence basins, which contain no nuclei except protons The characteristics of valence basins are determined by the number of core basins they share boundaries with, referred to as the valence basin synaptic order This classification leads to asynaptic, monosynaptic, disynaptic, and polysynaptic valence basins, where monosynaptic basins typically correspond to lone pair regions, while di- and polysynaptic basins signify chemical bonds This representation offers a distinct criterion for identifying multicentric bonds in molecular structures.
This perspective offers a complementary approach to traditional valence representation by shifting the focus from counting bonds around a central atom—typically limited to two-body interactions—to evaluating the connections from the 'glue' that binds the atoms together.
From a quantitative point of view a localization basin (core or valence) is characterized by its population, i.e the integrated one-electron densityρ( r )over the basin (equation 15) ¯
To calculate the variance of the basin population, we use the equation σ²(N¯; i) = ∫ i d³r₁ ∫ i d³r₂ P(r₁, r₂) - [N(¯i)]² + N(i), where N(i) represents the volume of the basin and P(r₁, r₂) is the spinless pair-electron density This variance can be expressed as a sum of contributions from other basins, highlighting the concept of covariance.
In equation 17, N(¯i)N(¯j) represents the expected number of electron pairs based on basin population, while N(¯i,j) denotes the actual number of pairs derived from integrating the pair-electron function over basins i and j The variance serves as a metric for the quantum mechanical uncertainty of basin population, reflecting electron delocalization, while pair covariance measures the correlation of population fluctuations between two specific basins Within the AIM framework, atomic localization and delocalization indices, λ(A) and δ(A, B), are defined by equations 18 and 19, where λ(A) equals ¯N(A) minus the variance σ²(N¯; A), and δ(A, B) is calculated as 2N(¯A)N(¯B) minus 2.
The AIM delocalization indices, often referred to as bond orders, can be applied to any partition in direct space, which is the approach taken in this study Utilizing the ELF method, the core population variance and core valence delocalization indices help determine if a specific core influences the synaptic order of a neighboring valence basin For instance, in the LiF molecule, the variances for the C(Li) and C(F) basins are measured at 0.09 and 0.38, respectively.
The localization domain concept is essential for understanding chemical properties, defined as a volume bounded by closed isosurfaces η(r) = f These domains can be classified as irreducible, surrounding a single attractor, or reducible, containing multiple attractors Typically, irreducible domains are filled volumes, while reducible ones may be filled, hollow, or donut-shaped As the value of η(r) increases, reducible domains can split into smaller domains with fewer attractors Critical points, known as localization nodes, mark the turning points where this reduction occurs By arranging these nodes in order of increasing η(r), one can create tree diagrams that illustrate the hierarchy of localization basins A core basin is included in the synaptic order of valence basins if a localization function value results in a hollow volume domain that encompasses the core domain within it.
Before proceeding further with bridging theELF with the key properties of mono- halophenols, we pause briefly to analyse analytically the vector gradient field ofELF.
Vector gradient field ∇ r η(r)
Applying the gradient toη( r )defined by equation 14, we derive equation 20,
[(t−t W ) 2 +t TF 2 ] 2 [(t−t W )∇t TF −t TF ∇(t−t W )] (20) where∇ r ≡ ∇ for short Assuming molecular orbitals to be real valued, equation 20 is then easily transformed to equation 21, ρ 1/3 [(t−t W ) 2 +t TF 2 ] 2
Therefore, we finally obtain equation 22 260, 261 ,
[(t−t W ) 2 +t TF 2 ] 2 ρ 10/3 ∇(J 2 /ρ 8 /3) (22) whereJ 2 is given by equation 23 270, 271 ,
Summarizing, the vector field∇η( r )of theELFvanishes at thoser∈R 3 which obey the conditiont ( r )=t W [ρ( r )] or equation 24,
For one purpose let us rewrite equation 23 as equation 25,
The electron transition current density between the ith and jth molecular orbitals is represented by the equation J 2 = N i