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Acta Physico-Chimica Sinca  2015, Vol. 31 Issue (11): 2057-2063    DOI: 10.3866/PKU.WHXB201509183
THEORETICALAND COMPUTATIONAL CHEMISTRY     
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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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 wordsRényi entropy      Tsallis entropy      Onicescu information energy      Shannon entropy      Density functional reactivity theory     
Received: 12 August 2015      Published: 18 September 2015
MSC2000:  O641  
Fund:  the National Natural Science Foundation of China(21503076);Aid Program for Science and TechnologyInnovative Research Team in Higher Educational Institutions of Hunan Province, China(湘教通[2012]318号)
Corresponding Authors: Shu-Bin. LIU,Chun-Ying. RONG     E-mail: shubin@email.unc.edu;rongchunying@aliyun.com
Cite this article:

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.

URL:

http://www.whxb.pku.edu.cn/10.3866/PKU.WHXB201509183     OR     http://www.whxb.pku.edu.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.
 
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  
 
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
 
 
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