Graphs and thermodynamics

Graphs are used in many fields of chemistry for codification and model purposes. One of these fields is widely known under the acronym QSAR/QSPR, i.e., quantitative structure–activity/structure–property relationships. In chemical graph studies directed graphs, known as digraphs, i.e., graphs with a preferred direction, have mainly been used to codify chemical reaction networks. Actually, digraphs, especially directed acyclic graphs together with simple graphs, can be used to draw a metalanguage of thermodynamics that codifies rules and properties which can be used to automatically derive many well-known, and less-known, thermodynamic relationships.     

Domain‐Averaged Exchange‐Correlation Energies as a Physical Underpinning for Chemical Graphs

A novel solution to the problem of assigning a molecular graph to a collection of nuclei (i.e. how to draw a molecular structure) is presented. Molecules are universally understood as a set of nuclei linked by bonds, but establishing which nuclei are bonded and which are not is still an empirical matter. Our approach borrows techniques from quantum chemical topology, which showed for the first time the construction of chemical graphs from wave functions, shifting the focus on energetics. This new focus resolves issues surrounding previous topological analyses, in which domain‐averaged exchange‐correlation energies (V_xc), quantities defined in real space between each possible atom pair, hold the key. Exponential decay of V_xc in non‐metallic systems as the intercenter distance increases guarantees a well‐defined hierarchy for all possible V_xc values in a molecule. Herein, we show that extracting the set of atom pairs that display the largest V_xc values in the hierarchy is equivalent to retrieving the molecular graph itself. Notably, domain‐averaged exchange‐correlation energies are transferable, and they can be used to calculate bond strengths. Fine‐grained details resulted to be related to simple stereoelectronic effects. These ideas are demonstrated in a set of simple pilot molecules.     

On Some Structural Properties of Fullerene Graphs

We show how some important structural properties of general fullerene graphs follow from the recently proved fact that all fullerene graphs are cyclically 4-edge connected. These properties, in turn, give us upper and lower bounds for various graph invariants. In particular, we establish the best currently known lower bound for the number of perfect matchings in fullerene graphs.     

Graphs with maximum connectivity index

Let G be a graph and d_v the degree (=number of first neighbors) of its vertex v. The connectivity index of G is χ=∑ (d_ud_v)^−1/2, with the summation ranging over all pairs of adjacent vertices of G. In a previous paper (Comput. Chem. 23 (1999) 469), by applying a heuristic combinatorial optimization algorithm, the structure of chemical trees possessing extremal (maximum and minimum) values of χ was determined. It was demonstrated that the path P_n is the n-vertex tree with maximum χ-value. We now offer an alternative approach to finding (molecular) graphs with maximum χ, from which the extremality of P_n follows as a special case. By eliminating a flaw in the earlier proof, we demonstrate that there exist cases when χ does not provide a correct measure of branching: we find pairs of trees T, T′, such that T is more branched than T′, but χ(T)>χ(T′). The smallest such examples are trees with 36 vertices in which one vertex has degree 31.     

Bonding Energies and Structural Study of Alkylaluminium

Neutral aluminium alkyls are well known to act as ethylene oligomerization and polymerization catalysts and cocatalysts.On the basis of the full optimization of alkylaluminium compounds with Gaussian 98 program package at the B3LYP/6-31G** level,the selected structures and bonding energies were investigated extensively.The geometries and bonding energies of AlR3(R = H,CH3,C2H5,C3H7,C4H9) and Al(C2H5)2R'(R' = C2H5,C3H7,C4H9,C5H11,C6H13) were investigated extensively,and we found that,along with the prolongation of carbon chains the terminal C-C bond is shortened gradually until to a constant value of about 0.1532 nm in C4H9;and the bonding energy almost remains constant.The dative bonding of C2H4 to Al(C2H5)3,whose bonding energy is only 15.30 kJ/mol,is very weak.     

Ab initio Study on Ionization Energies of 1-Methyl-hypoxanthine

Six low-lying tautomers of 1-methyl-hypoxanthine have been studied at the B3LYP/aug-cc-pVDZ level. Two tautomers N7H and N9H with the comparable energies are far more stable than the others. The vertical ionization energies of the tautomers calculated with ab initio electron propagator theory in the P3/aug-cc-pVDZ approximation are in agreement with the experimental data from photoelectron spectroscopy. According to the calculated relative energies and the comparison between the simulated and the experimental photoelectron spectra, it demonstrates that there are at least two tautomers of 1-methyl-hypoxanthine in the gas-phase experiments.     

Accurate solutions for the spiked oscillators

The one‐dimensional harmonic oscillator potential plus a term of the form λ / x^α is known as the spiked oscillator potential; it constitutes a very interesting system because of its difficulties to accept perturbative and variational solutions for certain regimes of the α parameter. By the use of a numerical method, we obtain accurate energy eigenvalues and eigenfunctions for a wide range of λ values and a few α values. The accuracy of the present results is by much higher than previously published results. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Proper Choice of XC Functionals and Calculations of Fluorescence-Emitting Energies for Coumarin Derivatives

A scheme of time-dependent density functional theory (TDDFT) combined with single-excitation configuration interaction (CIS) approach was employed to make a detailed investigation of the emitting energy for fifteen well-known coumarin derivatives. The results showed that the predicted emitting energies as well as the absorption ones were dominated mainly by the exchange-correlation (XC) functional to be used. So long as a functional is properly chosen, the experimental emitting energy of most derivatives can be accurately reproduced within 0.16 eV by a calculation at the TDDFT/6-31G(d)//CIS/3-21G(d) theoretical level. It was found that, nevertheless, the hybrid functional, B3LYP, well predicted the absorption energies for all the fifteen coumarin derivatives but none of the functionals could work equally well for the emitting energy calculations. Two pure functionals, OLYP and BLYP, yield good emitting energies for the 7-aminocoumarins or derivatives with a N atom connected to 7-position, which exhibit inconspicuous charge transfer (CT) in their excited states, whereas the B3LYP hybrid functional, with 20% Hartree-Fock (HF) exchange energy, performs significantly better than OLYP and BLYP for those 3-substituted coumarins with larger CT in excited states. Thus, in comparison with the absorption energies, the selection of proper functionals for the emitting energy calculations becomes more complex. In all probability, it is effective and doable to choose an XC-functional with alterable fraction of HF exchange energy according to the composition and structure characteristics of molecule.     

香豆素衍生物的荧光发射能计算及XC泛函的合理选择

采用含时密度泛函理论(TDDFT)与单激发组态相互作用(CIS)处理相结合的计算方案对香豆素系列15种已知荧光化合物的发射能进行了系统考察. 结果表明, 发射能与吸收能一样, 其计算值主要取决于交换-相关(XC)泛函的选择. 只要泛函选用得当, 在使用较小基组的TDDFT/6-31G(d)//CIS/3-21G(d)理论水平上即可使绝大部分化合物的实验发射能在精度达0.16 eV以内得以重现. 与吸收能计算不同的是, 无法选用单一的一种泛函来对全系列化合物的发射能作出满意的理论预测. 激发态无明显电荷转移的、7位上有氨(或胺)基取代或有氮原子相连的化合物, 其适用泛函为不含Hartree-Fock(HF)交换能的纯泛函OLYP和BLYP. 而激发态发生较大程度电荷转移的、3 位上有共轭取代基的衍生物, 其适用泛函则为含20%的HF交换成分的混合泛函B3LYP. 因此, 发射能计算中的XC泛函选择, 应同时考虑取代基团效应以及激发态的电子结构特征. 其中, 发射能计算值受XC泛函中HF交换能比例的影响十分敏感. 文中还对激发能计算中的溶剂效应校正方案和激发态几何优化精度的影响进行了讨论.     

Unicyclic Graphs with Minimal Energy

If G is a graph and ₁,₂,..., _n are its eigenvalues, then the energy of G is defined as E(G)=|₁|+|₂|++| _n |. Let S _n ³ be the graph obtained from the star graph with n vertices by adding an edge. In this paper we prove that S _n ³ is the unique minimal energy graph among all unicyclic graphs with n vertices (n6).     

14种结合自由能评价函数的比较

采用LigandFit作为构象采样工具，以230个蛋白质-配体复合物组成的预测集，系统地比较了14种自由能评价函数（Ligscore1、Ligscore2、Plp1、Plp2、Jain、Pmf、Ludi1、Ludi2、Ludi3、D-score、Pmf-score、G-score、Chemscore以及Xscore）对蛋白质和小分子之间的结合模式以及结合自由能的预测能力. Plp1、Plp2、G-score、Pmf和Xscore在预测测试集结合自由能时得到的分数同实验测定的结合自由能的线性相关系数大于50%. 在识别配体分子实验结合构象的能力方面， 选择测试构象与实际构象间的位置均方根偏差rmsd≤0.20 nm作为评价标准，14种评价函数的成功率从46%到77%不等，其中Ligscore1、Ligscore2、Plp1、Plp2以及Xscore的成功率都在70%以上. 将评价函数中的2个或者3个组合得到一组共同评价函数可以进一步提高实验构象的预测能力， 其预测成功率可以达到80%. 实验表明Xscore、Plp1和Plp2在对接和评价方面都得到较好的结果.     

分子内结构环境对解离能的影响

为了理解化学键的这一结构效应, 本文对具有相同化学键而分子内结构环境不同的系列分子进行了计算研究, 讨论了化学键结构环境对解离能的影响.     

卤素双原子分子部分电子态的完全振动能谱和离解能

用代数方法(AM)可以获得双原子分子包含最高振动能级在内的所有高阶振动能级的完全振动能谱; 基于Leroy和Bernstein的能级表达式, 研究了卤族元素双原子分子Cl2-A’3∏(2u)、Br2-X1∑+g和I2-0+u电子态的完全振动能谱和离解能, 得到的理论结果与实验符合得很好.     

Evaluating the Free Energies of Solvation and Electronic Structures of Lithium‐Ion Battery Electrolytes

Adaptive biasing force molecular dynamics simulations and density functional theory calculations were performed to understand the interaction of Li⁺ with pure carbonates and ethylene carbonate (EC)‐based binary mixtures. The most favorable Li carbonate cluster configurations obtained from molecular dynamics simulations were subjected to detailed structural and thermochemistry calculations on the basis of the M06‐2X/6‐311++G(d,p) level of theory. We report the ranking of these electrolytes on the basis of the free energies of Li‐ion solvation in carbonates and EC‐based mixtures. A strong local tetrahedral order involving four carbonates around the Li⁺ was seen in the first solvation shell. Thermochemistry calculations revealed that the enthalpy of solvation and the Gibbs free energy of solvation of the Li⁺ ion with carbonates are negative and suggested the ion–carbonate complexation process to be exothermic and spontaneous. Natural bond orbital analysis indicated that Li⁺ interacts with the lone pairs of electrons on the carbonyl oxygen atom in the primary solvation sphere. These interactions lead to an increase in the carbonyl (C=O) bond lengths, as evidenced by a redshift in the vibrational frequencies [ν(C=O)] and a decrease in the electron density values at the C=O bond critical points in the primary solvation sphere. Quantum theory of atoms in molecules, localized molecular orbital energy decomposition analysis (LMO‐EDA), and noncovalent interaction plots revealed the electrostatic nature of the Li⁺ ion interactions with the carbonyl oxygen atoms in these complexes. On the basis of LMO‐EDA, the strongest attractive interaction in these complexes was found to be the electrostatic interaction followed by polarization, dispersion, and exchange interactions. Overall, our calculations predicted EC and a binary mixture of EC/dimethyl carbonate to be appropriate electrolytes for Li‐ion batteries, which complies with experiments and other theoretical results.     

InSb的Li替位形成能的从头计算

采用基于平面波展开的第一原理赝势法计算了锑化铟在锂替位到铟位置时的各种情况下的形成能与电子结构,讨论了锂替位的体积变化,电荷分布、能带结构及电子态密度等性质.结果表明,对于闪锌矿结构的InSb,锂的各种替位形成能大致在每个锂原子-2.2eV左右.该结果表明,不可能在嵌入初期Li插入到间隙位置之前发生替位反应,与实验结果一致.     

紫外光照射对皮胶原热降解活化能的影响

将皮胶原经不同时间的紫外光照射,采用热重法(TG)和微分热重分析(DTG)研究了照射前后皮胶原的热降解行为,用Horowitz-Metzger法和Coats-Redfern法计算了其热降解活化能,得到了未照射皮胶原和经过不同紫外线照射时间处理后的皮胶原的热降解活化能。研究发现,在较短时间(0~4h)的照射后,皮胶原的热降解活化能略有增大;在较长时间(8~64h)的照射后,皮胶原的热降解活化能大幅降低。这可能是由于在紫外光照射的过程中,皮胶原分子链发生了以交联为主和以断链为主的复杂光化学反应。     

基团贡献法估算线性烷基醇分子体系电子相关能

在本文中,提出了极性基团电子相关能贡献的定义,并在MP2-OPT2/6-311++G(d)水平上计算了CH₃(CH₂) _mOH( m=0-4)体系中HO-、CH₃-和-CH₂-基团电子相关能贡献值。计算结果表明,在CH₃(CH₂) _mOH( m=0-4)体系中端基HO-、CH₃-基团电子相关能贡献值 E_corr(HO-)和 E_corr(cH₃-)的数值随着 m的增加而逐渐减小。同一体系中a -CH₂-基团电子相关能贡献值大于其它-CH₂-基团电子相关能贡献值,在CH₃(CH₂) _mOH( m=1-4)体系中,距离端基HO-基团越远的-CH₂-基团其电子相关能贡献值越小;通过计算结果可以推断,在CH₃(CH₂) _mOH体系中随着 m的逐渐增加,相对远离端基HO-的-CH₂-基团的电子相关能贡献值表现出收敛趋向并将趋于不变,此-CH₂-基团可看作一个标准的亚甲基而且其 E_corr(-cH₂-)的数值在CH₃(CH₂) _mOH体系中具有传递性。在MP2-OPT2/6-311++G(d)水平上对CH₃(CH₂) _mOH( m=2-4)体系的计算结果和应用Gaussian 98程序在MP2/6-311++G(d)//HF/6-311++G(d)水平上对CH₃(CH₂) _mOH( m=2-7)体系的计算结果均表明,体系总电子相关能与（ m-1）呈 中 m是体系中亚甲乙烯基的数值。     

Fibonacci relations. On the computation of some counting polynomials of very large graphs

A definition of a set of Fibonacci graphs is introduced which allows construction of several counting polynomials of very large graphs quite easily using a pencil-and-a-paper approach. These polynomials include matching, sextet, independence, Aihara and Hosoya polynomials. Certain combinatorial properties of Kekulé counts of benzenoid hydrocarbons are given. A relation to a new topological function that counts the cardinality of graph topology [23] is given.Dedicated to Professor Oskar E. Polansky for his enthusiastic support, participation and promotion of chemical graph theory.