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物理化学学报  2015, Vol. 31 Issue (11): 2057-2063    DOI: 10.3866/PKU.WHXB201509183
理论与计算化学     
密度泛函活性理论中的Rényi熵, Tsallis熵和Onicescu信息能
刘述斌1,2,*(),荣春英1,*(),吴泽民1,卢天3
1 湖南师范大学化学化工学院,资源精细化与先进材料湖南省高校重点实验室,化学生物学及中药分析教育部重点实验室,长沙410081
2 Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, USA
3 北京科音自然科学研究中心,北京100022
Rényi Entropy, Tsallis Entropy and Onicescu Information Energy in Density Functional Reactivity Theory
Shu-Bin. LIU1,2,*(),Chun-Ying. RONG1,*(),Ze-Min. WU1,Tian. LU3
1 Key Laboratory of Sustainable Resources Processing and Advanced Materials of Hunan Province College, Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha 410081, P. R. China
2 Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, USA
3 Beijing Kein Research Center for Natural Sciences, Beijing 100022, P. R. China
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摘要:

根据密度泛函理论,分子的电子密度确定了该体系基态下的所有性质,其中包括结构和反应活性.如何运用电子密度泛函有效地预测分子反应活性仍然是一个有待解决的难题.密度泛函活性理论(DFRT)倾力打造这样一个理论和概念架构,使得运用电子密度以及相关变量准确地预测分子的反应特性成为可能.信息理论方法的香农熵和费舍尔信息就是这样的密度泛函,研究表明,它们均可作为反应活性的有效描述符.本文将在DFRT框架中介绍和引进三个密切相关的描述符, Rényi熵、Tsallis熵和Onicescu信息能.我们准确地计算了它们在一些中性原子和分子中的数值并讨论了它们随电子数量和电子总能量的变化规律.此外,以第二阶Onicescu信息能为例,在分子和分子中的原子两个层面上,系统地考察了其随乙烷二面角旋转的变化模式.这些新慨念的引入将为我们深入洞察和预测分子的结构和反应活性提供额外的描述工具.

关键词: Rényi熵Tsallis熵Onicescu信息能香农熵密度泛函活性理论    
Abstract:

Density functional theory dictates that the electron density determines everything in a molecular system's ground state, including its structure and reactivity properties. However, little is known about how to use density functionals to predict molecular reactivity. Density functional reactivity theory is an effort to fill this gap: it is a theoretical and conceptual framework through which electron-related functionals can be used to accurately predict structure and reactivity. Such density functionals include quantities from the information-theoretic approach, such as Shannon entropy and Fisher information, which have shown great potential as reactivity descriptors. In this work, we introduce three closely related quantities: Rényi entropy, Tsallis entropy, and Onicescu information energy. We evaluated these quantities for a number of neutral atoms and molecules, revealing their scaling properties with respect to electronic energy and the total number of electrons. In addition, using the example of second-order Onicescu information energy, we examined how its patterns change with the angle of dihedral rotation of an ethane molecule at both the molecular level and atoms-in-molecules level. Using these quantities as additional reactivity descriptors, researchers can more accurately predict the structure and reactivity of molecular systems.

Key words: Rényi entropy    Tsallis entropy    Onicescu information energy    Shannon entropy    Density functional reactivity theory
收稿日期: 2015-08-12 出版日期: 2015-09-18
中图分类号:  O641  
基金资助: 国家自然科学基金(21503076);湖南省高校科技创新团队支持计划(湘教通[2012]318号)
通讯作者: 刘述斌,荣春英     E-mail: shubin@email.unc.edu;rongchunying@aliyun.com
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引用本文:

刘述斌,荣春英,吴泽民,卢天. 密度泛函活性理论中的Rényi熵, Tsallis熵和Onicescu信息能[J]. 物理化学学报, 2015, 31(11): 2057-2063.

Shu-Bin. LIU,Chun-Ying. RONG,Ze-Min. WU,Tian. LU. Rényi Entropy, Tsallis Entropy and Onicescu Information Energy in Density Functional Reactivity Theory. Acta Physico-Chimica Sinca, 2015, 31(11): 2057-2063.

链接本文:

http://www.whxb.pku.edu.cn/CN/10.3866/PKU.WHXB201509183        http://www.whxb.pku.edu.cn/CN/Y2015/V31/I11/2057

Atom Na E2 E3 R2 R3 T2 T3 Ea
H 1 0.039 0.002 1.41 1.21 0.96 0.50 –0.50
He 2 0.763 0.389 0.12 0.05 0.24 0.11 –2.90
Li 3 3.139 6.170 –0.50 –0.55 –2.14 –5.67 –7.48
Be 4 8.40 42.76 –0.92 –0.97 –7.40 –42.26 –14.66
C 6 31.87 580.9 –1.50 –1.53 –30.87 –580.37 –37.84
N 7 52.69 1538.8 –1.72 –1.74 –51.69 –1538.28 –54.58
O 8 81.65 3567.3 –1.91 –1.93 –80.65 –3566.82 –75.06
F 9 120.3 7455.5 –2.08 –2.09 –119.3 –7455.0 –99.72
Ne 10 170.5 14377.0 –2.23 –2.23 –169.5 –14376.5 –128.92
Na 11 234.0 26194.0 –2.37 –2.36 –233.0 –26193.5 –162.24
Mg 12 313.0 45493.0 –2.50 –2.48 –312.0 –45492.5 –200.05
Si 14 523.3 120384.6 –2.72 –2.69 –522.3 –120384.1 –289.35
P 15 658.1 185814.7 –2.82 –2.79 –657.1 –185814.2 –341.24
S 16 815.3 278831.0 –2.91 –2.87 –814.3 –278830.5 –398.10
Cl 17 996.0 407976.6 –3.00 –2.96 –995.0 –407976.1 –460.13
Ar 18 1203.0 583504.5 –3.08 –3.03 –1202.0 –583504.0 –527.52
K 19 1440.9 820114.8 –3.16 –3.11 –1439.9 –820114.3 –599.90
Ca 20 1708.0 1132761 –3.23 –3.18 –1707.0 –1132761 –677.56
Cr 24 3094.7 3531757 –3.49 –3.42 –3093.7 –3531757 –1044.32
Mn 25 3530.3 4544622 –3.55 –3.48 –3529.3 –4544622 –1150.77
Co 27 4533.3 7324860 –3.66 –3.58 –4532.3 –7324860 –1382.53
Ni 28 5105.7 9187318 –3.71 –3.63 –5104.7 –9187318 –1508.14
Zn 30 6386.7 14073009 –3.81 –3.72 –6385.7 –14073009 –1779.26
Ge 32 7889.8 20976835 –3.90 –3.81 –7888.8 –20976835 –2076.90
As 33 8725.5 25385817 –3.94 –3.85 –8724.5 –25385816 –2235.80
Se 34 9621.0 30546397 –3.98 –3.89 –9620.0 –30546397 –2401.50
Br 35 10577.7 36545294 –4.02 –3.93 –10576.7 –36545293 –2574.13
Kr 36 11603.3 43529405 –4.06 –3.97 –11602.3 –43529404 –2753.80
                 
R2(N)b 0.834 0.653 0.806 0.806 0.834 0.653  
R2(E)b   0.986 0.874 0.558 0.558 0.986 0.874  
Onicescu information energy of orders 2 and 3, E2 and E3; Rényi entropy of orders 2 and 3, R2 and R3; and Tsallis entropy of orders 2 and 3, T2 and T3, for a list of neutral atoms. aN is the total number of atoms; E is the total electronic energy. bR2 is the correlation coefficients of these quantities with respect to both N and E.
Table 1  Atomic values of new information-theoretic quantities (units are in atomic unit) obtained with the total electron density
Atom N $E\!_2^\sigma$ $E\!_3^\sigma$ $R\!_2^\sigma$ $R\!_3^\sigma$ $T\!\; _2^\sigma$ $T\!\; _3^\sigma$ E
H 1 0.04 0.00 1.41 1.21 0.96 0.50 –0.50
He 2 0.19 0.10 0.72 0.51 0.81 0.31 –2.90
Li 3 0.35 0.46 0.46 0.17 0.65 0.11 –7.48
Be 4 0.53 1.34 0.28 –0.06 0.47 –0.33 –14.66
C 6 0.89 5.38 0.05 –0.37 0.11 –2.36 –37.84
N 7 1.08 8.97 –0.03 –0.48 –0.08 –4.15 –54.58
O 8 1.28 13.93 –0.11 –0.57 –0.28 –6.63 –75.06
F 9 1.49 20.45 –0.17 –0.66 –0.49 –9.89 –99.72
Ne 10 1.70 28.75 –0.23 –0.73 –0.70 –14.04 –128.92
Na 11 1.93 39.36 –0.29 –0.80 –0.93 –19.35 –162.24
Mg 12 2.17 52.65 –0.34 –0.86 –1.17 –25.99 –200.05
Si 14 2.67 87.74 –0.43 –0.97 –1.67 –43.54 –289.35
P 15 2.92 110.11 –0.47 –1.02 –1.92 –54.72 –341.24
S 16 3.18 136.15 –0.50 –1.07 –2.18 –67.74 –398.10
Cl 17 3.45 166.08 –0.54 –1.11 –2.45 –82.71 –460.13
Ar 18 3.71 200.10 –0.57 –1.15 –2.71 –99.72 –527.52
K 19 3.99 239.14 –0.60 –1.19 –2.99 –119.23 –599.90
Ca 20 4.27 283.19 –0.63 –1.23 –3.27 –141.26 –677.56
Cr 24 5.37 510.96 –0.73 –1.35 –4.37 –255.15 –1044.32
Mn 25 5.65 581.71 –0.75 –1.38 –4.65 –290.52 –1150.77
Co 27 6.22 744.28 –0.79 –1.44 –5.22 –371.81 –1382.53
Ni 28 6.51 837.04 –0.81 –1.46 –5.51 –418.19 –1508.14
Zn 30 7.10 1042.45 –0.85 –1.51 –6.10 –520.89 –1779.26
Ge 32 7.70 1280.32 –0.89 –1.55 –6.70 –639.83 –2076.90
As 33 8.01 1412.80 –0.90 –1.58 –7.01 –706.06 –2235.80
Se 34 8.32 1554.37 –0.92 –1.60 –7.32 –776.85 –2401.50
Br 35 8.63 1704.74 –0.94 –1.62 –7.63 –852.04 –2574.13
Kr 36 8.95 1865.97 –0.95 –1.64 –7.95 –932.65 –2753.80
                 
R2(N)   0.994 0.838 0.773 0.788 0.994 0.838  
R2(E)   0.946 0.988 0.526 0.539 0.946 0.988  
Table 2  Atomic values of new information-theoretic quantities (units are in atomic unit) obtained with the shape density for neutral atoms up to Kr
R1R2R3 Na E2 E3 R2 R3 T2 T3 E
HHH 28 270.90 15285.70 –2.433 –2.243 –269.90 –15285.20 –239.55
CH3HH 36 302.58 15846.16 –2.481 –2.250 –301.58 –15845.66 –278.85
C2H5HH 44 334.23 16404.65 –2.524 –2.258 –333.23 –16404.15 –318.15
CH3CH3H 44 334.26 16407.22 –2.524 –2.258 –333.26 –16406.72 –318.16
C2H5CH3H 52 365.92 16965.94 –2.563 –2.265 –364.92 –16965.44 –357.46
C3H7HH 52 365.90 16963.84 –2.563 –2.265 –364.90 –16963.34 –357.45
CH3CH3CH3 52 365.94 16966.97 –2.563 –2.265 –364.94 –16966.47 –357.47
C2H5C2H5H 60 397.58 17524.38 –2.599 –2.272 –396.58 –17523.88 –396.76
C2H5CH3CH3 60 397.60 17525.81 –2.599 –2.272 –396.60 –17525.31 –396.77
C3H7CH3H 60 397.58 17525.07 –2.599 –2.272 –396.58 –17524.57 –396.76
C4H9HH 60 397.57 17524.58 –2.599 –2.272 –396.57 –17524.08 –396.75
C2H5C2H5CH3 68 429.26 18084.94 –2.633 –2.279 –428.26 –18084.44 –436.07
C3H7C2H5H 68 429.25 18084.21 –2.633 –2.279 –428.25 –18083.71 –436.06
C3H7CH3CH3 68 429.27 18085.16 –2.633 –2.279 –428.27 –18084.66 –436.07
C2H5C2H5C2H5 76 460.92 18643.97 –2.664 –2.286 –459.92 –18643.47 –475.36
C3H7C2H5CH3 76 460.93 18644.35 –2.664 –2.286 –459.93 –18643.85 –475.36
C3H7C3H7H 76 460.92 18643.53 –2.664 –2.286 –459.92 –18643.03 –475.35
C3H7C2H5C2H5 84 492.60 19203.48 –2.692 –2.292 –491.60 –19202.98 –514.66
C3H7C3H7CH3 84 492.60 19203.53 –2.692 –2.292 –491.60 –19203.03 –514.66
C3H7C3H7C2H5 92 524.27 19762.76 –2.720 –2.298 –523.27 –19762.26 –553.96
C3H7C3H7C3H7 100 555.94 20322.10 –2.745 –2.304 –554.94 –20321.60 –593.26
                 
R2(E)   1.000 1.000 0.988 0.998 1.000 1.000  
R2(N)   1.000 1.000 0.988 0.998 1.000 1.000  
Onicescu information energy of orders 2 and 3, E2 and E3; Rényi entropy of orders 2 and 3, R2 and R3; and Tsallis entropy of orders 2 and 3, T2 and T3, for a total of 21 molecular systems of the same kind with the general formula of R1R2R3C-F
Table 3  Molecular values of new information-theoretic quantities (units are in atomic unit) obtained with the total electron density
Fig 1  Profiles of information-theoretic quantities as a function of the ∠ HCCH dihedral angle for the ethane molecule (a) the Onicescu information energy of orders 2 (E2) of C2H6; (b) the Shannon entropy of C2H6; (c) atomic E2 values for carbon atoms in C2H6; (d) atomic E2 values for hydrogen atoms in C2H6
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