Density Functionals of Chemical Bonding
Abstract
:1. Introduction
2. Primary Density Functional Theory Concepts
2.1. HohenbergKohn theorems
2.2. Optimized energyelectronegativity connection
 It starts with a trial electronic density (20) satisfying the Ncontingency conditions (14) and (15);
 With trial density the effective potential (31) containing exchange and correlation is calculated;
 With the set of functions ${\left\{{\varphi}_{i}(\text{r})\right\}}_{i=\overline{1,N}}$ the new density (20) is recalculated;
 The procedure is repeated until the difference between two consecutive densities approaches zero;
 Once the last condition is achieved one retains the last set ${\left\{{\varphi}_{i}(\text{r}),{\mu}_{i}={\chi}_{i}\right\}}_{i=\overline{1,N}}$;
 The electronegativity orbital observed contributions are summed up from (33) with the expression:$$\underset{i}{\overset{N}{\Sigma}}\u3008{\chi}_{i}\u3009=\underset{i}{\overset{N}{\Sigma}}\int {n}_{i}{\phi}_{i}^{*}(\text{r})\left[\frac{1}{2}{\nabla}^{2}+{V}_{eff}(\text{r})\right]{\phi}_{i}(\text{r})d\text{r}={T}_{s}[\rho ]+\int {V}_{eff}\rho (\text{r})d\text{r};$$
 Replacing in (34) the uniform kinetic energy, T_{s} [ρ] from the general relation (21) the density functional of the total energy for the Nelectronic system will take the final figure [9, 24]:$$E[\rho ]=\underset{i}{\overset{N}{\Sigma}}\u3008{\chi}_{i}\u3009\frac{1}{2}\int \int \frac{\rho ({\text{r}}_{1})\rho ({\text{r}}_{2})}{\left{\text{r}}_{12}\right}d{\text{r}}_{1}d{\text{r}}_{2}+\left\{{E}_{xc}[\rho ]\int {V}_{xc}(\text{r})\rho (\text{r})d\text{r}\right\}.$$
3. Electronic Localization Problem
3.1. From global functional to localization function. Localization in solids
3.2. Localization in atoms and molecules
 Choosing the basis of the atomic functions [65]:$$\begin{array}{c}{f}_{1}^{Li}(r)=8.863248rexp(2.698r),\\ {f}_{2}^{Li}(r)=0.369721{r}^{5/2}exp(0.797r)\end{array}$$
 Generating the orthonormal orbital eigenwaves, here according with the GramSchmidt algorithm among shells and subshells:$$\begin{array}{c}{\phi}_{1s}^{Li}(r)={f}_{1}^{Li}(r),\hspace{0.17em}{\phi}_{2p}^{Li}(r)={f}_{2}^{Li}(r),\\ {\phi}_{2s}^{Li}(r)={C}_{2s}^{Li}\left[{\phi}_{2p}^{Li}(r)\alpha {\phi}_{1s}^{Li}(r)\right]\\ =1.2226exp(2.698r)+0.373222{r}^{5/2}exp(0.797r)\end{array}$$
 Generating the working overall electronic density$${\rho}_{Li}(r)=2{\left[{\phi}_{1s}^{Li}(r)\right]}^{2}+{\left[{\phi}_{2s}^{Li}(r)\right]}^{2}$$
4. Popular Energetic Density Functionals
4.1. Density functionals of kinetic energy
 the Gaussian resummation uses:$$\text{\Gamma}(\text{r},s)\cong {\text{\Gamma}}_{G}(\text{r},s)=\text{exp}\left(\frac{{s}^{2}}{\beta (\text{r})}\right)$$
 the trigonometric (uniform gas) approximation looks like:$$\text{\Gamma}(\text{r},s)\cong {\text{\Gamma}}_{T}(\text{r},s)=9\frac{{(sinttcost)}^{2}}{{t}^{6}},t=s\sqrt{\frac{5}{\beta (\text{r})}}$$
 in Gaussian resummation:$${T}_{G}^{LDA}=\frac{3\pi}{{2}^{5/3}}\int {\rho}^{5/3}(\text{r})d\text{r},$$
 whereas in trigonometric approximation$${T}_{TF}^{LDA}=\frac{3}{10}{\left(3{\pi}^{2}\right)}^{2/3}\int {\rho}^{5/3}(\text{r})d\text{r}$$
4.2. Density functionals of exchange energy
 in Gaussian resummation:$${K}_{G}^{LDA}=\frac{1}{{2}^{1/3}}\int {\rho}^{4/3}(\text{r})d\text{r};$$
 and in trigonometric approximation (recovering the Dirac formula):$${K}_{G}^{LDA}=\frac{3}{4}{\left(\frac{3}{\pi}\right)}^{1/3}\int {\rho}^{4/3}(\text{r})d\text{r}.$$
4.3. Density functionals of correlation energy
 The gradient corrected correlation functional reads as [102, 104]:$$\begin{array}{c}{E}_{c}^{GC}=\int d\text{r}{\epsilon}_{c}[{\rho}_{\uparrow},{\rho}_{\downarrow}]\rho (\text{r})+\int d\text{r}{B}_{c}^{p}{[{\rho}_{\uparrow},{\rho}_{\downarrow}]}_{c[\rho ]=\sqrt{2\pi}/4{({6\pi}^{2})}^{4/3},f=0.17}\nabla \rho (\text{r}){}^{2}+\\ +9\frac{\pi}{4{(6{\pi}^{2})}^{4/3}}{(0.17)}^{2}\int d\text{r}\left(\nabla {\rho}_{\uparrow}{}^{2}{\rho}_{\uparrow}^{4/3}+\nabla {\rho}_{\downarrow}{}^{2}{\rho}_{\downarrow}^{4/3}\right)\end{array}$$
 The Lee, Yang, and Parr (LYP) functional within ColleSalvetti approximation unfolds like [105]:$$\begin{array}{c}{E}_{c}^{LYP}={a}_{c}{b}_{c}\int d\text{r}\gamma (\text{r})\xi (\text{r})\left(\underset{\sigma}{\Sigma}{\rho}_{\sigma}(\text{r})\underset{i}{\Sigma}\nabla {\phi}_{i\sigma}(\text{r}){}^{2}\frac{1}{4}\underset{\sigma}{\Sigma}{\rho}_{\sigma}(\text{r})\text{\Delta}{\rho}_{\sigma}(\text{r})\frac{1}{4}\nabla \rho (\text{r}){}^{2}+\frac{1}{4}\rho (\text{r})\text{\Delta}\rho (\text{r})\right)\\ {a}_{c}\int d\text{r}\frac{\gamma (\text{r})}{\eta (\text{r})}\rho (\text{r})\end{array}$$
 The openshell (OS) case provides the functional [98]:$${E}_{c}^{OS}=\int d\text{r}\frac{{a}_{s}\rho (\text{r})+{b}_{s}\nabla \rho (\text{r})\rho {(\text{r})}^{1/3}}{{c}_{s}+{d}_{s}\left(\nabla {\rho}_{\uparrow}{\rho}_{\uparrow}^{4/3}+\nabla {\rho}_{\downarrow}{\rho}_{\downarrow}^{4/3}\right)+{r}_{s}}\sqrt{1{\zeta}^{2}}$$
 Finally, Perdew and Zunger (PZ) recommend the working functional [106]:$${E}_{c}^{PZ0}[\rho ]=\int d\text{r}\rho (\text{r})\frac{{\alpha}_{p}}{1+{\beta}_{1p}\sqrt{{r}_{s}}+{\beta}_{2p}{r}_{s}}$$
4.4. Density functionals of exchangecorrelation energy
5. Testing (χ, η) Quadratic Dependency Among Several Energetic Density Functionals
5.1. Proof of the E=E(χ,η) quadratic dependency
5.2. Atomic and molecular analysis of the energetic quadratic bilinear (χ, η) dependency
 the degree of correlation itself between the employed energy functional and the couple of electronegativitychemical hardness structural indices; this is measured by the standard correlation factor [113]:$$r=\sqrt{1\frac{\underset{i}{\Sigma}{({y}_{iINPUT}{y}_{iFIT})}^{2}}{\underset{i}{\Sigma}{({y}_{iINPUT}\overline{{y}_{iINPUT}})}^{2}}}$$
 the degree of parabolic dependency by checking whether the chemical hardness coefficient (c) is the square of the electronegativity coefficient (b) thus giving the opportunity of introducing the socalled sigmapi reactivity index$${\sigma}_{\pi}=sign(b)\frac{c}{{b}^{2}}\stackrel{\text{parabolic}E=E(\chi ,\eta )}{\to}1$$
 the correlation energy appears to provide acceptable parabolic shapes in both atomic and molecular cases, with better bilinear regression for molecular analysis, while strongly depending on the electronegativity and chemical hardness atomic models and scales;
 the kinetic energy, while displaying poor parabolic shape at atomic level behaves with negative chemical hardness in molecular systems, probably due the positive contribution in bonding that compete with stabilization (localization) of the electrons within internuclear basin;
 exchange and exchangecorrelation functionals reveal similar reactive (parabolic) efficiency as well as close bivariate regression correlation factors for both atomic and molecular cases, leaving with the impression that the exchange contribution is dominant in exchangecorrelation functionals since cancelling somehow the behavior of the correlation part of the functional.
 overall, the total energy, although with correlation factors in the range of its components’ regressions does not fit with parabolic reactive theoretical prescription (189), at least for present employed set of atoms and molecules.
6. Conclusions
Acknowledgements
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Atoms  Kinetic energy  Exchange energy  Correlation energy  

T_{exact}♣  T_{0}♣  T_{0}+T_{2}♣  T_{Padé}♣  K_{exact}♦  K_{0}♥  K^{B88}♥  E_{c}^{exact}♦  E_{c}^{(139)}♠  E_{c}•  
He  2.86168  2.56054  2.87850  2.87639  −1.0260  −0.884  −1.025  −0.0425  −0.0215  −0.0681 
Li  7.43273  6.70062  7.50504  7.44941  −1.7812  −1.538  −1.775  −0.0454  −0.0486  −0.0815 
Be  14.5730  13.1286  14.6466  14.4223  −2.6669  −2.312  −2.658  −0.0945  −0.0820  −0.1192 
B  24.5291  22.0720  24.5228  24.2089  −3.7438  −3.272  −3.728  −0.1247  −0.1197  −0.1625 
C  37.6886  34.0144  37.5988  37.2533  −5.0444  −4.459  −5.032  −0.1566  −0.1609  −0.2091 
N  54.4009  49.4771  54.3852  54.0643  −6.5971  −5.893  −6.589  −0.1850  −0.2050  −0.2567 
O  74.8094  67.8965  74.3573  74.1625  −8.1752  −7.342  −8.169  −0.2579  −0.2512  −0.3035 
F  99.4093  90.4598  98.6429  98.6959  −10.003  −9.052  −10.02  −0.332  −0.2996  −0.3510 
Ne  128.547  117.761  127.829  128.221  −12.108  −11.03  −12.14  −0.390  −0.3498  −0.3987 
Na  161.859  148.809  161.093  161.718  −14.017  −12.79  −14.03  −0.398  −0.3892  −0.4137 
Mg  199.614  184.017  198.749  199.578  −15.994  −14.61  −16.00  −0.443  −0.4351  −0.4491 
Al  241.877  223.443  240.868  242.008  −18.069  −16.53  −18.06  −0.480  −0.4809  −0.4863 
Si  288.854  267.315  287.659  289.139  −20.280  −18.59  −20.27  −0.521  −0.5308  −0.5308 
Cl  459.482  426.865  457.321  460.117  −27.512  −25.35  −27.49  −0.714  −0.6901  −0.6710 
Ar  526.817  490.017  524.289  527.617  −30.185  −27.86  −30.15  −0.787  −0.7459  −0.7190 
Atoms  ExchangeCorrelation energy  Total energy  

E_{xc}^{exact}♣  E_{xc}^{I(Xα)}♦  E_{xc}^{II(Xα)}♦  E_{xc}^{I(Wig)}♦  E_{xc}^{II(Wig)}♦  E_{tot}^{exact}♥  E_{tot}^{xc(RG)}♥  E_{tot}^{xc(LDA)}♥  E_{tot}^{BLYP}♠  E_{tot}^{PW91}♠  
He  −1.0685  −1.0604  −1.0566  −1.0633  −1.0654  −2.9042  −3.0317  −2.8601  −2.9071  −2.9000 
Li  −1.8266  −1.8048  −1.8134  −1.8093  −1.8108  −7.4781  −7.6473  −7.3704  −7.4827  −7.4742 
Be  −2.7614  −2.7260  −2.7522  −2.7325  −2.7342  −14.6675  −14.8911  −14.4966  −14.6615  −14.6479 
B  −3.8685  −3.8126  −3.8415  −3.8215  −3.8177  −24.6538  −24.9158  −24.4097  −24.6458  −24.6299 
C  −5.2010  −5.1127  −5.1338  −5.1248  −5.1121  −37.8163  −38.1305  −37.5095  −37.8430  −37.8265 
N  −6.7821  −6.6400  −6.6440  −6.6558  −6.6321  −54.4812  −54.8681  −54.1287  −54.5932  −54.5787 
O  −8.4331  −8.3599  −8.3405  −8.3796  −8.3450  −75.0271  −75.4597  −74.5979  −75.0786  −75.0543 
F  −10.325  −10.327  −10.277  −10.350  −10.305  −99.741  −100.235  −99.247  −99.7581  −99.7316 
Ne  −12.498  −12.551  −12.466  −12.579  −12.524  −128.937  −129.522  −128.403  −128.9730  −128.9466 
Na  −14.415  −14.462  −14.382  −14.488  −14.445  −162.257  −162.862  −161.624  −162.293  −162.265 
Mg  −16.437  −16.482  −16.424  −16.504  −16.484  −200.058  −200.705  −199.340  −200.093  −200.060 
Al  −18.549  −18.566  −18.542  −18.583  −18.593  −242.357  −243.028  −241.533  −242.380  −242.350 
Si  −20.801  −20.774  −20.791  −20.784  −20.830  −289.356  −290.063  −288.435  −289.388  −289.363 
Cl  −28.226  −28.115  −28.272  −28.092  −28.281  −460.196  −461.005  −458.963  −460.165  −460.147 
Ar  −30.972  −30.827  −31.037  −30.789  −31.035  −527.605  −528.452  −526.267  −527.551  −527.539 
Level  FiniteDifference  Functional  Semiclassical  

Atoms  χ_{FD}  η_{FD}  χ_{DFT}  η_{DFT}  χ_{SC}  η_{SC} 
He  0.45094  0.45866  1.21132  1.66189  0.57038  0.2172 
Li  0.11099  0.16134  0.15105  0.08784  0.00412  0.00334 
Be  0.12606  0.21794  0.44248  0.44579  0.00893  0.0047 
B  0.15656  0.14921  1.15362  1.34105  0.01526  0.00588 
C  0.22933  0.18339  2.76332  2.9695  0.02279  0.00684 
N  0.25616  0.27894  5.79566  4.91363  0.03139  0.0076 
O  0.27894  0.22566  10.6505  5.91694  0.04072  0.00816 
F  0.38221  0.25983  16.9129  4.37707  0.05061  0.00849 
Ne  0.39361  0.40132  23.7119  −0.08747  0.06079  0.00864 
Na  0.10290  0.10621  0.23153  0.18743  0.00011  0.00005 
Mg  0.09555  0.18339  0.49871  0.53142  0.00018  0.00007 
Al  0.11834  0.10327  1.04631  1.19882  0.00026  0.00008 
Si  0.17200  0.12606  2.10805  2.30724  0.00036  0.00009 
Cl  0.30577  0.17120  11.5766  7.7692  0.00074  0.00012 
Ar  0.28299  0.29806  17.8831  9.08857  0.00088  0.00013 
Molecules  Kinetic  Exchange  Correlation  Exch.corr.  Total energy  

T_{0}♣  T_{0}+T_{2}♣  K^{exact}♣  K^{PBE}♣  E_{c}^{VWN}♦  E_{c}^{GCP}♦  E_{xc}^{exact}♣  E_{xc}^{PBE}♣  E_{tot}^{BLYP}♥  E_{tot}^{TH}♥  
H_{2}  1.140  1.125  −0.657  −0.648  −95·10^{−3}  −47·10^{−3}  −0.698  −0.691  −1.169  −1.178 
LiH  7.978  8.003  −2.125  −2.105  −219·10^{−3}  −93·10^{−3}  −2.212  −2.188  −8.068  −8.070 
CH_{4}  40.050  40.141  −6.576  −6.536  −593·10^{−3}  −328·10^{−3}  −6.883  −6.836  −40.502  −40.515 
H_{2}O  76.150  75.477  −8.910  −8.917  −664·10^{−3}  −365·10^{−3}  −9.292  −9.241  −76.448  −76.433 
HF  100.137  99.242  −10.378  −10.385  −704·10^{−3}  −380·10^{−3}  −10.779  −10.720  −100.48  −100.455 
N_{2}  109.115  108.242  −13.094  −13.128  −945·10^{−3}  −506·10^{−3}  −13.665  −13.580  −109.559  −109.54 
O_{2}  149.843  148.369  −16.290  −16.358  −1110·10^{−3}  −599·10^{−3}  −16.958  −16.887  −150.384  −150.337 
F_{2}  198.892  196.729  −19.872  −19.951  −1302·10^{−3}  −697·10^{−3}  −20.661  −20.564  −199.599  −199.533 
Level Molecule s  FiniteDifference  Functional  Semiclassical  

χ_{FD}  η_{FD}  χ_{DFT}  η_{DFT}  χ_{SC}  η_{SC}  
H_{2}  0.26387  0.2370  0.26384  0.23704  0.26387  0.23705 
LiH  0.15626  0.192  0.19212  0.12818  0.00811  0.006596 
CH_{4}  0.25616  0.2239  0.32216  0.29051  0.08468  0.03064 
H_{2}O  0.26871  0.2331  0.39097  0.34859  0.09335  0.0229 
HF  0.3077  0.2479  0.51964  0.44974  0.08493  0.01639 
N_{2}  0.25616  0.2789  5.79566  4.91363  0.03139  0.00761 
O_{2}  0.27894  0.2257  10.6505  5.91694  0.04072  0.00816 
F_{2}  0.36898  0.2598  16.9129  4.37707  0.05061  0.00849 
Method of
 QSPR results
 Method of
 QSPR results
 

Energ y  (χ, η)  a  b  c  σ_{π}  r  Energy  (χ, η)  a  b  c  σ_{π}  r 
T_{exact}  FD  194.59  741.55  −951.38  −0.0017  0.33  E_{xc}^{exact}  FD  −15.30  −44.80  60.89  −0.0303  0.37 
DFT  51.73  2.09  31.51  7.2  0.62  DFT  −6.56  −0.19  −1.53  40.84  0.58  
SC  186.38
 −2315.7
 5147.4
 −0.001
 0.35
 SC  −13.76
 73.07
 −128.0
 −0.024
 0.38
 
T_{0}  FD  179.67  687.26  −881.14  −0.0019  0.33  E_{xc}^{I(Xα)}  FD  −15.28  −44.56  60.69  −0.0306  0.37 
DFT  46.99  1.91  29.41  8.07  0.62  DFT  −6.56  −0.2  −1.51  39.11  0.57  
SC  172.42
 −2185.3
 4874.1
 −0.001
 0.35
 SC  −13.72
 72.04
 −125.5
 −0.024
 0.38
 
T_{0}+T_{2}  FD  193.83  736.94  −946.29  −0.0017  0.33  E_{xc}^{II(Xα)}  FD  −15.28  −44.81  60.92  −0.0303  0.37 
DFT  51.59  2.07  31.36  7.33  0.62  DFT  −6.52  −0.19  −1.53  40.81  0.57  
SC  185.59
 −2312.1
 5142
 −0.001
 0.35
 SC  −13.75
 74.72
 −132.4
 −0.024
 0.38
 
T_{Padé}  FD  194.61  741.17  −951.43  −0.0017  0.33  E_{xc}^{I(Wig)}  FD  −15.29  −44.53  60.67  −0.0306  0.37 
DFT  51.54  2.08  31.58  7.3  0.62  DFT  −6.58  −0.2  −1.50  38.78  0.57  
SC  186.42
 −2334.4
 5197.4
 −0.001
 0.35
 SC  −13.72
 71.4
 −123.8
 −0.024
 0.38
 
K_{exact}  FD  −14.91  −43.62  59.38  −0.0312  0.37  E_{xc}^{II(Wig)}  FD  −15.30  −44.80  60.95  −0.0304  0.37 
DFT  −6.37  −0.19  −1.49  43.17  0.57  DFT  −6.54  −0.2  −1.52  39.98  0.57  
SC  −13.4
 72.74
 −128.78
 −0.0243
 0.38
 SC  −13.76
 74.1
 −130.7
 −0.024
 0.38
 
K_{0}  FD  −13.59  −40.29  54.69  −0.0337  0.37  E_{tot}^{exact}  FD  −194.98  −742.7  952.91  −0.0017  0.33 
DFT  −5.72  −0.17  −1.39  46.96  0.58  DFT  −51.91  −2.10  −31.54  7.15  0.62  
SC  −12.24
 68.77
 −123.42
 −0.026
 0.38
 SC  −186.73
 2316
 −5148
 −0.001
 0.35
 
K^{B88}  FD  −14.89  −43.61  59.33  −0.0312  0.37  E_{tot}^{xc(RG)}  FD  −195.55  −743.9  954.5  −0.0017  0.33 
DFT  −6.37  −0.19  −1.49  42.35  0.57  DFT  −52.27  −2.11  −31.56  7.1  0.62  
SC  −13.39
 71.88
 −126.54
 −0.0245
 0.38
 SC  −187.24
 2315.
 −5143.
 −0.001
 0.35
 
E_{c}^{exact}  FD  −0.39  −1.20  1.52  −1.055  0.38  E_{tot}^{xc(LDA)}  FD  −194.26  −740.7  950.13  −0.0017  0.33 
DFT  −0.19  −0.0075  −0.03  595.98  0.59  DFT  −51.59  −2.09  −31.47  7.18  0.62  
SC  −0.362
 0.232
 0.997
 18.528
 0.38
 SC  −186.1
 2313.
 −5142.
 −0.001
 0.35
 
E_{c}^{(139)}♠  FD  −0.40  −1.13  1.55  −1.207  0.39  E_{tot}^{BLYP}  FD  −194.99  −742.7  952.86  −0.0017  0.33 
DFT  −0.19  −0.0056  −0.03  1081.3  0.57  DFT  −51.94  −2.1  −31.53  7.15  0.62  
SC  −0.356
 0.851
 −0.563
 −0.778
 0.41
 SC  −186.74
 2315.
 −5145
 −0.001
 0.35
 
E_{c}•  FD  −0.41  −1.10  1.44  −1.1808  0.40  E_{tot}^{PW91}  FD  −194.97  −742.7  952.81  −0.0017  0.33 
DFT  −0.22  −0.0063  −0.03  735.5  0.59  DFT  −51.92  −2.1  −31.54  7.15  0.62  
SC  −0.374  −0.224  2.113  −42.24  0.42  SC  −186.7  2315.  −5145.  −0.001  0.35 
Method of
 QSPR results
 Method of
 QSPR results
 

Energy y  χ & η  a  b  c  σ_{π}  r  Energy y  χ & η  a  b  c  σ_{π}  r 
T_{0}  FD  −192.71  821.61  238.59  0.00035  0.77  T_{0}+T_{2}  FD  −190.58  811.48  237.96  0.0004  0.77 
DFT  40.446  8.125  4.501  0.068  0.89  DFT  40.199  8.012  4.497  0.07  0.89  
SC  82.14
 542.16
 −977.9
 −0.0033
 0.56
 SC  81.4
 537.74
 −969.47
 −0.003
 0.56
 
K^{exact}  FD  18.18  −70.1  −38.02  0.0077  0.74  K^{PBE}  FD  18.324  −70.48  −38.25  0.0077  0.74 
DFT  −5.234  −0.646  −0.805  1.93  0.89  DFT  −5.219  −0.65  −0.811  1.92  0.89  
SC  −9.78
 −48.27
 94.96
 −0.04
 0.61
 SC  −9.81
 −48.15
 95.
 −0.04
 0.61
 
E_{c}^{VWN}  FD  1.039  −4.06  −2.74  0.167  0.71  E_{c}^{GCP}  FD  0.59  −2.27  −1.496  0.289  0.72 
DFT  −0.423  −0.035  −0.06  47.83  0.87  DFT  −0.225  −0.019  −0.033  94.08  0.86  
SC  −0.71
 −3.13
 6.22
 −0.63
 0.64
 SC  −0.36  −2.1  3.7  −0.88  0.64  
E_{xc}^{exact}  FD  18.95  −72.63  −40.1  0.0076  0.74  E_{xc}^{PBE}  FD  18.85  −72.45  −39.62  0.0075  0.74 
DFT  −5.457  −0.668  −0.846  1.897  0.89  DFT  −5.421  −0.666  −0.84  1.898  0.89  
SC  −10.19  −50.16  98.71  −0.039  0.61  SC  −10.13  −49.97  98.28  −0.04  0.61  
E_{tot}^{BLYP}  FD  193.12
 −824.17
 −238.96
 0.00035
 0.77
 E_{tot}^{TH}  FD  193.06
 −823.8
 −239.04
 0.0004
 0.77

DFT  −40.67  −8.149  −4.514  0.068  0.89  DFT  −40.67  −8.145  −4.514  0.068  0.89  
SC  −82.48  −544.35  981.7  −0.003  0.56  SC  −82.46  −544.3  981.5  −0.003  0.56 
Share and Cite
Putz, M.V. Density Functionals of Chemical Bonding. Int. J. Mol. Sci. 2008, 9, 10501095. https://doi.org/10.3390/ijms9061050
Putz MV. Density Functionals of Chemical Bonding. International Journal of Molecular Sciences. 2008; 9(6):10501095. https://doi.org/10.3390/ijms9061050
Chicago/Turabian StylePutz, Mihai V. 2008. "Density Functionals of Chemical Bonding" International Journal of Molecular Sciences 9, no. 6: 10501095. https://doi.org/10.3390/ijms9061050