Density Functionals of Chemical Bonding
Abstract
:1. Introduction
2. Primary Density Functional Theory Concepts
2.1. Hohenberg-Kohn theorems
2.2. Optimized energy-electronegativity connection
- It starts with a trial electronic density (20) satisfying the N-contingency conditions (14) and (15);
- With trial density the effective potential (31) containing exchange and correlation is calculated;
- With the set of functions 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 ;
- The electronegativity orbital observed contributions are summed up from (33) with the expression:
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]:
- Generating the orthonormal orbital eigen-waves, here according with the Gram-Schmidt algorithm among shells and sub-shells:
- Generating the working overall electronic density
4. Popular Energetic Density Functionals
4.1. Density functionals of kinetic energy
- the Gaussian resummation uses:
- the trigonometric (uniform gas) approximation looks like:
- in Gaussian resummation:
- whereas in trigonometric approximation
4.2. Density functionals of exchange energy
- in Gaussian resummation:
- and in trigonometric approximation (recovering the Dirac formula):
4.3. Density functionals of correlation energy
- The open-shell (OS) case provides the functional [98]:
- Finally, Perdew and Zunger (PZ) recommend the working functional [106]:
4.4. Density functionals of exchange-correlation 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 electronegativity-chemical hardness structural indices; this is measured by the standard correlation factor [113]:
- 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 so-called sigma-pi reactivity index
- 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 exchange-correlation 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 exchange-correlation 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
References
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Atoms | Kinetic energy | Exchange energy | Correlation energy | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Texact♣ | T0♣ | T0+T2♣ | TPadé♣ | Kexact♦ | K0♥ | KB88♥ | Ecexact♦ | Ec(139)♠ | Ec• | |
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 | Exchange-Correlation energy | Total energy | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Excexact♣ | ExcI(Xα)♦ | ExcII(Xα)♦ | ExcI(Wig)♦ | ExcII(Wig)♦ | Etotexact♥ | Etotxc(RG)♥ | Etotxc(LDA)♥ | EtotBLYP♠ | EtotPW91♠ | |
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 | Finite-Difference | Functional | Semiclassical | |||
---|---|---|---|---|---|---|
Atoms | χFD | ηFD | χDFT | ηDFT | χSC | ηSC |
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 | |||||
---|---|---|---|---|---|---|---|---|---|---|
T0♣ | T0+T2♣ | Kexact♣ | KPBE♣ | EcVWN♦ | EcGCP♦ | Excexact♣ | ExcPBE♣ | EtotBLYP♥ | EtotTH♥ | |
H2 | 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 |
CH4 | 40.050 | 40.141 | −6.576 | −6.536 | −593·10−3 | −328·10−3 | −6.883 | −6.836 | −40.502 | −40.515 |
H2O | 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 |
N2 | 109.115 | 108.242 | −13.094 | −13.128 | −945·10−3 | −506·10−3 | −13.665 | −13.580 | −109.559 | −109.54 |
O2 | 149.843 | 148.369 | −16.290 | −16.358 | −1110·10−3 | −599·10−3 | −16.958 | −16.887 | −150.384 | −150.337 |
F2 | 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 | Finite-Difference | Functional | Semiclassical | |||
---|---|---|---|---|---|---|
χFD | ηFD | χDFT | ηDFT | χSC | ηSC | |
H2 | 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 |
CH4 | 0.25616 | 0.2239 | 0.32216 | 0.29051 | 0.08468 | 0.03064 |
H2O | 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 |
N2 | 0.25616 | 0.2789 | 5.79566 | 4.91363 | 0.03139 | 0.00761 |
O2 | 0.27894 | 0.2257 | 10.6505 | 5.91694 | 0.04072 | 0.00816 |
F2 | 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 | σπ | r | Energy | (χ, η) | a | b | c | σπ | r |
Texact | FD | 194.59 | 741.55 | −951.38 | −0.0017 | 0.33 | Excexact | 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
| ||
T0 | FD | 179.67 | 687.26 | −881.14 | −0.0019 | 0.33 | ExcI(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
| ||
T0+T2 | FD | 193.83 | 736.94 | −946.29 | −0.0017 | 0.33 | ExcII(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
| ||
TPadé | FD | 194.61 | 741.17 | −951.43 | −0.0017 | 0.33 | ExcI(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
| ||
Kexact | FD | −14.91 | −43.62 | 59.38 | −0.0312 | 0.37 | ExcII(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
| ||
K0 | FD | −13.59 | −40.29 | 54.69 | −0.0337 | 0.37 | Etotexact | 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
| ||
KB88 | FD | −14.89 | −43.61 | 59.33 | −0.0312 | 0.37 | Etotxc(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
| ||
Ecexact | FD | −0.39 | −1.20 | 1.52 | −1.055 | 0.38 | Etotxc(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
| ||
Ec(139)♠ | FD | −0.40 | −1.13 | 1.55 | −1.207 | 0.39 | EtotBLYP | 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
| ||
Ec• | FD | −0.41 | −1.10 | 1.44 | −1.1808 | 0.40 | EtotPW91 | 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 | σπ | r | Energy y | χ & η | a | b | c | σπ | r |
T0 | FD | −192.71 | 821.61 | 238.59 | 0.00035 | 0.77 | T0+T2 | 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
| ||
Kexact | FD | 18.18 | −70.1 | −38.02 | 0.0077 | 0.74 | KPBE | 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
| ||
EcVWN | FD | 1.039 | −4.06 | −2.74 | 0.167 | 0.71 | EcGCP | 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 | ||
Excexact | FD | 18.95 | −72.63 | −40.1 | 0.0076 | 0.74 | ExcPBE | 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 | ||
EtotBLYP | FD | 193.12
| −824.17
| −238.96
| 0.00035
| 0.77
| EtotTH | 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, 1050-1095. https://doi.org/10.3390/ijms9061050
Putz MV. Density Functionals of Chemical Bonding. International Journal of Molecular Sciences. 2008; 9(6):1050-1095. 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: 1050-1095. https://doi.org/10.3390/ijms9061050