<Text-field style="Heading 1" layout="Heading 1"><Font size="20">The effect of functional on simple organic molecules using Density functional theory</Font></Text-field><Text-field style="Heading 1" bold="false" size="12" layout="Heading 1" alignment="right"><Font size="12" bold="false">CHEM362
Ziqing Lin (Janey)</Font></Text-field><Text-field style="Heading 1" bold="false" size="12" layout="Heading 1"><Font size="16" bold="true">Introduction</Font><Font size="12" bold="false">
</Font><Font size="14" bold="false" encoding="UTF-8">Unlike wave mechanics that solve the Schrodinger equation by approximating wave function, density functional theory (DFT) uses density of electrons, thereby reducing the many body quantum mechanics to a one body problem. The challenge in quantum mechanics then is to find the approximation for true wavefunction, while in DFT, the goal is to find the best approximation of exchange-correlation functional Exc(\342\215\264) that enclose all the unknown parts. However, unlike wave mechanics where there is a clear systematic strategy to improve the approximate wave function, there is no such guideline in DFT. In fact, the quality of DFT calculation depends solely on the accuracy of the chosen exchange-correlation functional. Therefore, it is interesting to compare the performance of various functional like LDA, GGA and hybrid functional with experimental value.
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LDA works best in a neutral uniform electron gas system, imagine a system has N electrons, and volume V. If both of these properties are approaching infinite, the ratio N/V will be finite and remain constant. In real world settings, this would most likely be a metal with a perfect crystal of valence electron that has uniformly distributed background charge, like sodium. In fact LDA is mostly used to study properties of metal, graphite or weakly bounded molecular crystals. However, LDA fails to capture the characteristics of systems that deviate greatly from homogeneity. In fact, it usually overestimates binding in molecules and underestimates bond length.
To address the limitations of LDA, two classes of functionals: generalized gradient approximation (GGA) and Hybrid functional were developed. GGA, as the name suggests, not only depends on density but also on the density gradients to track the changes of density. It is found that GGA can improve the atomization energy calculated from LDA by a factor of 3-5. Hybrid functional mixes the exact HF exchange into the functional and thereby has improved the accuracy of the approximation1.
In summary GGA offers good correction for LDA, and Hybrid functional offers further improvement for DFT calculations. In 2003, Boese, Martin and Handy performed an evaluation on various functionals, to explore properties like bond length and ionization energy (IE). They have created a ranking for functionals for both GGA and hybrid functional2. Inspired by their work, this project would explore the performance of 7 different functionals: LDA, 2 GGA functionals, and 4 hybrid functional on three organic molecules: acetylene, ethylene, ethane are chosen, using ground state energy, and ionization energy (IE). </Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" bold="false" size="14" layout="Heading 1"><Font size="16" bold="true">Method
</Font><Font size="14" bold="true">Density functional theory (DFT) </Font><Font size="14" bold="false">
<Font encoding="UTF-8">Density functional theory (DFT) is known for its ability to reduce the many body quantum mechanics to a one body problem using the Hohenberg-Kohn theorem and Kohn-Sham theorem. Wave mechanics works with N electron wavefunction, which contains three spatial coordinates for every electron, making a total of 3N coordinates. DFT, on the other hand, solves the Schrodinger equation not with wavefunction, but using the density of electrons (\342\215\264). The density of electrons could be understood as the probability of finding an electron in space with three spatial coordinates. Thus, DFT greatly reduces the complexity of the problem. According to the first Hohenberg-Kohn theorem, there is a one to one mapping between electron density and ground state wavefunction of molecules. Energy could be expressed in the form of a single electron density, where T, E</Font>eN, Eee refer to kinetic energy, electron-nuclear energy and electron-electron repulsion energy </Font><Equation executable="true" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbW9HRiQ2MFEifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1GIzYoLUkjbW5HRiQ2J1EiMUYnRjJGNUY4RjtGMkY1LyUnaXRhbGljR0Y6RjgvRjxRJ2l0YWxpY0YnLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YyRjVGOEY7">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</Equation><Font size="14" bold="false">.
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GVzYnLUYjNictRmJxNidRIjRGJ0YvRjJGNUY4Ri9GMkY1RjhGL0YyRjVGOEZqb0ZickZqb0YvRjJGNUY4">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</Equation></Text-field><Text-field style="Heading 1" size="14" layout="Heading 1"><Font size="14" bold="false" encoding="UTF-8">Since the functional does not depend on the system, but is actually a universal functional that can be used for all atoms and molecules, the task to solve for the ground state energy now becomes to solve for universal functional F(\342\215\264) that solely depend on the density of electron.
</Font><Font size="14" bold="false">
<Font encoding="UTF-8">Other than the Hohenberg-Kohn theorem, Kohn-Sham theorem also plays an important role in DFT. Kohn-Sham theorem suggests that a system of non-interacting electrons shares the same density as a system of interacting electrons. The energy according to the Kohn-Sham theorem has the following form, where \317\225i(r), \317\265i, V(r) refer to the Kohn-Sham orbitals, Kohn-Sham energies, and Kohn-Sham potential </Font></Font><Equation executable="true" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSIyRidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSIyRidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4</Equation><Font size="14" bold="false">.
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">-</Font><Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtZnJhY0dGJDYoLUYjNiotSSNtbkdGJDYmUSIxRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Y4RjsvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRkMvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRkMvRj9RJ2l0YWxpY0YnLUYjNiktRjU2JlEiMkYnRjhGO0Y+RjhGO0ZBL0ZFUSxbMjAwLDAsMjAwXUYnRkdGSy8lLmxpbmV0aGlja25lc3NHRjcvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGWC8lKWJldmVsbGVkR0Y9RjhGO0Y+">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</Equation><Equation executable="false" style="Heading 1" input-equation="" display="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">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</Equation><Font size="14" bold="false">+V(<Font encoding="UTF-8">r) \317\225</Font>i(r) <Font encoding="UTF-8">=\317\265i \317\225i</Font>(r) (5)</Font><Font size="14">
<Font bold="false">The Kohn-Sham potential is a sum of attraction between nuclei and electrons, the coulomb potential, as well as the exchange-correlation potential </Font></Font><Equation executable="true" style="Heading 1" input-equation="" display="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">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</Equation><Font size="14">
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">Vs(r) = Vne(r) +</Font><Equation executable="false" style="Heading 1" input-equation="" display="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">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</Equation><Font size="14" bold="false">dr' +Vxc (r) (6)
<Font encoding="UTF-8">The Kohn-Sham DFT energy becomes, where the superscript on kinetic energy denotes single electron equations, J(\342\215\264) is the coulomb part, and Exc(\342\215\264) is the exchange-correlation functional :
</Font></Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false" encoding="UTF-8">E(\342\215\264) =</Font><Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2L1ErJkludGVncmFsO0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJRi9GMkY1">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2L1ErJkludGVncmFsO0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJRi9GMkY1</Equation><Font size="14" bold="false" encoding="UTF-8">vs(r)\342\215\264(r)dr + TS(\342\215\264) +J(\342\215\264) + Exc(\342\215\264) (7)
</Font><Font size="14" bold="false">
<Font encoding="UTF-8">In HK DFT, the challenge is to find a universal functional, the challenge in Kohn-Sham DFT, however is to find the exact Exc(\342\215\264) that would provide the exact energy. However, similar to HK DFT, finding the exact Exc(\342\215\264) is very computationally expensive, and thereby, approximation are used. In fact, the major difference in various DFT methods arises from the choice of Exc(\342\215\264). Many functionals are developed to study different systems, two of the most well-known functional ones are LDA and GGA. </Font></Font><Font size="14">
</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 1" layout="Heading 1"> <Font size="14">Functionals</Font>
<Font size="14">Local Density Approximation (LDA)<Font bold="false">
<Font encoding="UTF-8">LDA stands for local density approximation which evaluates universal function as if the local density arises from homogeneous electron gas. The exchange-correlation functional is shown below, where xc(\342\215\264(r)) is the exchange-correlation energy per particle in homogeneous electron gas </Font></Font></Font><Equation executable="true" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSIzRidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSIzRidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4</Equation><Font size="14" bold="false">:
</Font><Equation executable="true" style="Heading 1" input-equation="" display="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">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</Equation><Font size="14" bold="false"> = </Font><Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2L1ErJkludGVncmFsO0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJRi9GMkY1">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2L1ErJkludGVncmFsO0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZJRi9GMkY1</Equation><Font size="14" bold="false" encoding="UTF-8">\342\215\264(r)\317\265xc(\342\215\264(r))dr (8)
</Font><Font size="14" bold="false">
<Font encoding="UTF-8">\317\265xc(\342\215\264) can be further break down into pure exchange part x(\342\215\264) and pure correlation part c(\342\215\264). The form of the exchange part is found by Slater and originally derived by Bloch and Dirac:
</Font></Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">= -</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Equation executable="false" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSZtcm9vdEdGJDYkLUYjNilGKy1GIzYnRistSSZtZnJhY0dGJDYoLUYjNi0tSSNtbkdGJDYmUSIzRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi9RIn5GJ0Y/RkJGRS8lJmZlbmNlR0ZELyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmZuLUYsNidRKCYjOTA3NjtGJ0Y/RkIvJSdpdGFsaWNHUSV0cnVlRicvRkZRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiotRiw2J1EickYnRj9GQkZcb0Zfb0Y/RkJGXG8vJStmb3JlZ3JvdW5kR1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRl5vLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Zeb0Zfb0Y/RkJGRUY/RkJGXG9GaW9GXHBGXnBGX28tRiM2KS1GLDYnUSNwaUYnRj9GQi9GXW9GREZFRj9GQkZcby9Gam9RLFsyMDAsMCwyMDBdRidGXHBGX28vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl1xLyUpYmV2ZWxsZWRHRkRGP0ZCRkVGK0Y/RkJGXG9GX28tRiM2KUY7Rj9GQkZcb0ZmcEZccEZfb0YrRj9GQkZF">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSZtcm9vdEdGJDYkLUYjNilGKy1GIzYnRistSSZtZnJhY0dGJDYoLUYjNi0tSSNtbkdGJDYmUSIzRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi9RIn5GJ0Y/RkJGRS8lJmZlbmNlR0ZELyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmZuLUYsNidRKCYjOTA3NjtGJ0Y/RkIvJSdpdGFsaWNHUSV0cnVlRicvRkZRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiotRiw2J1EickYnRj9GQkZcb0Zfb0Y/RkJGXG8vJStmb3JlZ3JvdW5kR1ErWzAsMTYwLDgwXUYnLyUscGxhY2Vob2xkZXJHRl5vLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR0Zeb0Zfb0Y/RkJGRUY/RkJGXG9GaW9GXHBGXnBGX28tRiM2KS1GLDYnUSNwaUYnRj9GQi9GXW9GREZFRj9GQkZcby9Gam9RLFsyMDAsMCwyMDBdRidGXHBGX28vJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl1xLyUpYmV2ZWxsZWRHRkRGP0ZCRkVGK0Y/RkJGXG9GX28tRiM2KUY7Rj9GQkZcb0ZmcEZccEZfb0YrRj9GQkZF</Equation><Font size="14" bold="false">(9)
<Font encoding="UTF-8">On the other hand, the explicit expression of correlation part \317\265c(\342\215\264) is not yet known, yet, it could be simulated accurately via quantum monte carlo</Font>.
<Font encoding="UTF-8">The electron density can be also expressed in an unrestricted manner by expressing the electron density \342\215\264 with two spin densities:</Font> </Font><Equation executable="true" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2KFEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjovJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNigtRi82KFEiYUYnRjJGNUY4RjtGPUYyRjVGOEY7Rj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjJGNUY7L0Y+USdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2KFEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjovJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNigtRi82KFEiYUYnRjJGNUY4RjtGPUYyRjVGOEY7Rj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjJGNUY7L0Y+USdub3JtYWxGJw==</Equation><Font size="14" bold="false"> and </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImJGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImJGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn</Equation><Font size="14" bold="false">, as shown below.
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false" encoding="UTF-8">\342\215\264(r) =</Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImFGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImFGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn</Equation><Font size="14" bold="false">(r)+ </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImJGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImJGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn</Equation><Font size="14" bold="false">(r) (10)
The increase of variables introduce a little changes on equation 8, which now becomes:
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZfcS1JI21vR0YkNi9RIn5GJy8lJXNpemVHUSMxNEYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRistSSVtc3VwR0YkNiUtSSVtc3ViR0YkNiYtSSNtaUdGJDYmUSJFRidGLy8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRictRiM2Jy1GUzYmUSN4Y0YnRi9GVkZZRi9GVi9GM1EqSGVhZGluZ34xRidGWS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictRiM2Jy1GUzYmUSRMREFGJ0YvRlZGWUYvRlZGam5GWS8lMXN1cGVyc2NyaXB0c2hpZnRHRl5vLUYsNi5RIihGJ0YvRjUvRjlGWEY7L0Y+RlhGP0ZBRkNGRS9GSFEsMC4xNjY2NjY3ZW1GJy9GS0ZfcC1GUDYmLUZTNidRKCYjOTA3NjtGJ0YvRlZGMkZZLUYjNictRlM2J1EiYUYnRi9GVkYyRllGLy8lK2V4ZWN1dGFibGVHRjpGam5GNUZcb0Zfb0YvRmpuRjU=</Equation><Font size="14" bold="false"> , </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImJGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImJGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn</Equation><Font size="14" bold="false">) = </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LlErJkludGVncmFsO0YnLyUlc2l6ZUdRIzE0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkdGLy8lK2V4ZWN1dGFibGVHRjcvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJ0Yy">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW9HRiQ2LlErJkludGVncmFsO0YnLyUlc2l6ZUdRIzE0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHUSV0cnVlRicvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkdGLy8lK2V4ZWN1dGFibGVHRjcvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJ0Yy</Equation><Font size="14" bold="false" encoding="UTF-8">\342\215\264(r)\317\265xc(</Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImFGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImFGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn</Equation><Font size="14" bold="false">(r) ,</Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImJGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYnLUYsNiZRImJGJ0YvRjJGNUYvLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSpIZWFkaW5nfjFGJy9GNlEnbm9ybWFsRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn</Equation><Font size="14" bold="false">(r) )dr (11)
By doing so, the local density approximation can have more flexibility, and thereby increase the accuracy when dealing with open shell systems where the number of and electrons are not equal.
</Font><Font size="14">Generalized gradient approximation (GGA) <Font bold="false">
In order to capture the non-homogeneous part of density, generalized gradient approximation (GGA) was developed. GGA, as the name suggests, uses gradient of the density to track the changes of density. The exchange-correlation functional is </Font></Font><Equation executable="true" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSI0RidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSI0RidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4</Equation><Font size="14" bold="false">:
</Font><Equation executable="true" style="2D Math" input-equation="" 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size="14" bold="false"> = </Font><Equation executable="true" style="2D Math" input-equation="" 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executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JS1GLDYnUSJhRidGL0YyRjVGOEYvL0Y5USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JS1GLDYnUSJhRidGL0YyRjVGOEYvL0Y5USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=</Equation><Font size="14" bold="false"> , </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JS1GLDYnUSJiRidGL0YyRjVGOEYvL0Y5USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2J1EoJiM5MDc2O0YnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JS1GLDYnUSJiRidGL0YyRjVGOEYvL0Y5USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=</Equation><Font size="14" bold="false">, </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbW9HRiQ2LlEoJm5hYmxhO0YnLyUlc2l6ZUdRIzE0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjExMTExMTFlbUYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2J1EoJiM5MDc2O0YnRi8vJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0YzUSdpdGFsaWNGJy1GIzYlLUZONidRImFGJ0YvRlFGVEZXRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictRiw2L1EifkYnRi9GVEYyRjVGOEY6RjxGPkZARkJGRC9GSEZGLUYsNi9RIixGJ0YvRlRGMkY1L0Y5RlNGOkY8Rj5GQEZCRkQvRkhRLDAuMzMzMzMzM2VtRidGXm8tRiw2L0YuRi9GVEYyRjVGOEY6RjxGPkZARkJGREZHLUZLNiZGTS1GIzYnLUZONidRImJGJ0YvRlFGVEZXRi9GUUZURldGaG5GW29GXm8tRiw2L1EiKUYnRi9GVEYyL0Y2RlNGOC9GO0ZTRjxGPkZARkIvRkVRLDAuMTY2NjY2N2VtRicvRkhGZ3BGL0Yy">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</Equation><Font size="14" bold="false">dr (12)
Similar to LDA, the exchange-correlation functional can also be split into exchange and correlation contributions. The exchange contribution is written below, where s is the reduced density gradient for spin :
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false"> = </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtSSVtc3ViR0YkNiYtRiw2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiZRInhGJ0Y4RjtGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictRiM2JS1GLDYmUSRMREFGJ0Y4RjtGPkY4L0Y/USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHRkYtSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSgmIzkwNzY7RidGOEY7Rj5GOEZPRjhGT0Y4Rk8=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtSSVtc3ViR0YkNiYtRiw2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiZRInhGJ0Y4RjtGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictRiM2JS1GLDYmUSRMREFGJ0Y4RjtGPkY4L0Y/USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHRkYtSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSgmIzkwNzY7RidGOEY7Rj5GOEZPRjhGT0Y4Rk8=</Equation><Font size="14" bold="false">-</Font><Equation executable="false" style="Heading 1" input-equation="" display="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">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</Equation><Font size="14" bold="false" encoding="UTF-8">F(s\320\261)</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">(r)dr (13)
The denominator of s offers corrections for small density regions.
</Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZqcC1JI21pR0YkNiNRIUYnLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRjNGOQ==">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</Equation><Font size="14" bold="false">s =</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">(14)
Notice that the function F is not specified in equation 10. Many approaches have been developed to mathematically define the function F, however, it is important to note that these approaches are not inspired by physical reason. The various approaches can be divided into two major classes: one is based on GGA exchange functional developed by Becke, the other class is derived by Perdew. </Font><Equation executable="true" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2KFEiRkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjovJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNigtRi82KFEiQkYnRjJGNUY4RjtGPUYyRjVGOEY7Rj0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNUY7L0Y+USdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2KFEiRkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjovJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNigtRi82KFEiQkYnRjJGNUY4RjtGPUYyRjVGOEY7Rj0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGNUY7L0Y+USdub3JtYWxGJw==</Equation><Font size="14" bold="false">is an example for the first class, with the functional define as follow, where is an empirical parameter:
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">=</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">(15)
FP86is an example for the second class, the function is written below:
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">=</Font><Equation executable="false" style="Heading 1" input-equation="" display="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">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</Equation><Font size="14" bold="false">(16)
The correlation contribution for GGA ,on the other hand, is even more complicated than the exchange correlation and thus can not be represented by a single explicit expression.
</Font><Font size="14">Hybrid functionals: <Font bold="false">
Recall that the exchange and correlation functional </Font></Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYpLUYsNiZRI1hDRidGL0YyRjVGLy8lJWJvbGRHRjQvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKkhlYWRpbmd+MUYnL0Y2USVib2xkRicvJStmb250d2VpZ2h0R0ZHLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYpLUYsNiZRI1hDRidGL0YyRjVGLy8lJWJvbGRHRjQvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKkhlYWRpbmd+MUYnL0Y2USVib2xkRicvJStmb250d2VpZ2h0R0ZHLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJw==</Equation><Font size="14" bold="false"> are composed of an exchange part </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYpLUYsNiZRIlhGJ0YvRjJGNUYvLyUlYm9sZEdGNC8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EqSGVhZGluZ34xRicvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkcvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYpLUYsNiZRIlhGJ0YvRjJGNUYvLyUlYm9sZEdGNC8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EqSGVhZGluZ34xRicvRjZRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRkcvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0Yn</Equation><Font size="14" bold="false"> and a correlation part </Font><Equation executable="true" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2KFEiRUYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjovJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNigtRi82KFEiQ0YnRjJGNUY4RjtGPUYyRjVGOEY7Rj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjJGNUY7L0Y+USdub3JtYWxGJw==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYmLUkjbWlHRiQ2KFEiRUYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjovJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNigtRi82KFEiQ0YnRjJGNUY4RjtGPUYyRjVGOEY7Rj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRjJGNUY7L0Y+USdub3JtYWxGJw==</Equation><Font size="14" bold="false">. In fact the absolute value of Ex is much greater than Ec and thereby if the </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUYsNiZRIlhGJ0YvRjJGNUYvL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUYsNiZRIlhGJ0YvRjJGNUYvL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=</Equation><Font size="14" bold="false"> is determined correctly, the accuracy of DFT calculation will be high. However, as shown in the previous discussion of exchange functional of LDA and GGA, accurate expression for </Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUYsNiZRIlhGJ0YvRjJGNUYvL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=">LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0dGJDYlLUYsNiZRIlhGJ0YvRjJGNUYvL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRic=</Equation><Font size="14" bold="false"> is hard to obtain. On the other hand, the exact exchange energy can be easily calculated with the following equation from HF </Font><Equation executable="true" style="Heading 1" input-equation="" display="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">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</Equation><Font size="14" bold="false">:
</Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZqcC1JI21vR0YkNi9RIn5GJy8lJXNpemVHUSMxNEYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSUYrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGKy1JJW1zdXBHRiQ2JS1JJW1zdWJHRiQ2Ji1JI21pR0YkNiZRIkVGJ0YvLyUnaXRhbGljR1EldHJ1ZUYnL0Y2USdpdGFsaWNGJy1GUzYmUSJ4RidGL0ZWRlkvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnLUYjNiUtRlM2JlEjSEZGJ0YvRlZGWUYvRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZqbkYvRjU=">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</Equation><Font size="14" bold="false">=</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">dr1dr2 (17)
<Font encoding="UTF-8">Wouldn\342\200\231t it be more accurate to combine the HF exchange with correlation part from DFT to obtain the exchange-correlation function. As a matter of fact, a new group of functionals called Hybrid functional is developed using this strategy. The exchange-correlation functional is expressed below:
</Font></Font><Equation executable="true" style="Heading 1" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZicS1JI21vR0YkNjFRIn5GJy8lJXNpemVHUSMxNEYnLyUlYm9sZEdRJmZhbHNlRicvJStleGVjdXRhYmxlR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZORitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrRitGK0YrLUklbXN1YkdGJDYmLUkjbWlHRiQ2KVEiRUYnRi9GMi8lJ2l0YWxpY0dGN0Y1RjgvRjxRJ2l0YWxpY0YnLUYjNigtRlU2KFEjWENGJ0YvRjJGWEY1RlpGL0YyRlhGNUZaLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJ0YvRjJGNUY7</Equation><Font size="14" bold="false"> =</Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtSSVtc3ViR0YkNiYtRiw2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiZRInhGJ0Y4RjtGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictRiM2JS1GLDYmUSNIRkYnRjhGO0Y+RjgvRj9RJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdGRkY4Rk8=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSVtc3VwR0YkNiUtSSVtc3ViR0YkNiYtRiw2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiZRInhGJ0Y4RjtGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictRiM2JS1GLDYmUSNIRkYnRjhGO0Y+RjgvRj9RJ25vcm1hbEYnLyUxc3VwZXJzY3JpcHRzaGlmdEdGRkY4Rk8=</Equation><Font size="14" bold="false">+</Font><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1cEdGJDYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEiRUYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRjI2JlEiQ0YnRjVGOEY7RjVGOEY7LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvSSttc2VtYW50aWNzR0YkUSdhdG9taWNGJy1GIzYlLUYyNiZRI0tTRidGNUY4RjtGNS9GPFEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZFLUkjbW9HRiQ2LlEifkYnRjVGTi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGWC8lKXN0cmV0Y2h5R0ZYLyUqc3ltbWV0cmljR0ZYLyUobGFyZ2VvcEdGWC8lLm1vdmFibGVsaW1pdHNHRlgvJSdhY2NlbnRHRlgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0Zhb0ZSRlJGNUZO">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</Equation><Font size="14" bold="false">(18)
The above idea will give promising results if applied to atoms, however, the situation of molecules is more complicated and is beyond the scope of this project.
</Font><Font size="12" bold="false">
</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRidGKy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==<Text-field style="Heading 1" size="14" layout="Heading 1"><Font size="14">Functional Ranking according to Boese, Martin and Handy (2003)
<Font bold="false">Boese, Martin and Handy have created a functional ladder for both GGA and hybrid functionals. For GGA functional, the following error ranking is found, where the lowest functional appears at the beginning </Font></Font><Equation executable="true" style="Heading 1" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSIyRidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSIyRidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4</Equation><Font size="14" bold="false">.
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false">HCTH < OLYP < BPW91 < BLYP < mPWPW91 < BP86 < PWPW91 < PBE. (19)
For hybrid functional, the following rank is found:
</Font><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation><Font size="14" bold="false" encoding="UTF-8"> \317\204-HCTH < B97-1 <\317\204-HCTH < VSXC \342\210\274 B97-2 \342\210\274 B3LYP \342\210\274 B98 < PBE1PBE < PKZB (20)
The functional LDA\357\274\233BP86, PBE from GGA functional \357\274\214and PBE0, B3LYP, B97-1, and B97-2 from hybrid functional are chosen for this project to explore the ground state and ionization energy of three organic molecules: acetylene, ethylene, ethane.</Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRidGKy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJStleGVjdXRhYmxlR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==<Text-field style="Heading 1" layout="Heading 1"><Font size="16">Applications</Font></Text-field>JSFHAfter loading the Quantum Chemistry package, we explore the ground-state energy and ionization energy with various functional using DFT.
<Text-field style="Heading 2" layout="Heading 2">Quantum Chemistry</Text-field>We set the number of Digits to be used in computations to 15 and load the Quantum Chemistry package using Maple's with command.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(*
c2h2m :=HartreeFock(c2h2);
DensityPlot3D(c2h2,c2h2m);
ch4m :=HartreeFock(ch4);
DensityPlot3D(ch4,ch4m);
c2h6m :=HartreeFock(c2h6);
DensityPlot3D(c2h6,c2h6m);
*)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRidGKy8lJXNpemVHUSMxNkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn<Text-field style="Heading 2" layout="Heading 2">Define molecules</Text-field>
First, we define the geometry, which we obtained from the MolecularGeometry 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Electron Density plot
Then we compare the electron density between three organc molecules. Afterwards, we compute the energy from DFT with the DensityFunctional command. We plug in the various exchange-correlation (xc) functional and the 6-31g basis set 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Ground state energyLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUkjbWlHRiQ2JlEmYzJoMmdGJy8lJXNpemVHUSMxNkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RKiZjb2xvbmVxO0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZALyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRk8tRiw2JlEoY29udmVydEYnRi9GMkY1LUkobWZlbmNlZEdGJDYlLUYjNi4tRlY2JS1GIzYnLUYsNiZRJmMyaDJkRidGL0YyRjUtRlY2Jy1GIzYkLUYsNiZRJmVfdG90RidGL0YyRjVGPEYvRjwvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGLy8lK2V4ZWN1dGFibGVHRkBGPEYvRjwtRjk2LlEnJnNkb3Q7RidGL0Y8Rj5GQUZDRkVGR0ZJRksvRk5RJjAuMGVtRicvRlFGXnAtRiw2JlEmVW5pdHNGJ0YvRjJGNS1GVjYnLUYjNiQtRiw2JlElVW5pdEYnRi9GMkY1RjxGL0Y8RmJvRmVvLUZWNiUtRiM2JC1GLDYmUShoYXJ0cmVlRidGL0YyRjVGPEYvRjwtRjk2LlEiLEYnRi9GPEY+L0ZCRjRGQ0ZFRkdGSUZLRl1wL0ZRUSwwLjMzMzMzMzNlbUYnLUYsNiZRJnVuaXRzRidGL0YyRjVGYXEtRiw2JlEjZVZGJ0YvRjJGNUYvRmhvRjxGL0Y8LUY5Ni5RIjtGJ0YvRjxGPkZkcUZDRkVGR0ZJRktGXXBGUC1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0ZecC8lJmRlcHRoR0Zlci8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYmUSZjMmg0Z0YnRi9GMkY1RjhGUi1GVjYlLUYjNi4tRlY2JS1GIzYoLUY5Ni5RIn5GJ0YvRjxGPkZBRkNGRUZHRklGS0ZdcEZfcC1GLDYmUSZjMmg0ZEYnRi9GMkY1LUZWNictRiM2JkZfb0YvRmhvRjxGL0Y8RmJvRmVvRi9GaG9GPEYvRjxGam9GYHBGY3BGanBGYXFGZ3FGYXFGanFGL0Zob0Y8Ri9GPEZdckZgci1GLDYmUSZjMmg2Z0YnRi9GMkY1RjhGUi1GVjYlLUYjNi4tRlY2JS1GIzYnLUYsNiZRJmMyaDZkRidGL0YyRjVGXnRGL0Zob0Y8Ri9GPEZqb0ZgcEZjcEZqcEZhcUZncUZhcUZqcUYvRmhvRjxGL0Y8Rl1yRmByLUZhcjYmRmNyRmZyRmhyL0Zbc1ElYXV0b0YnLUYsNiNRIUYnRi9GaG9GPA==<Text-field style="Heading 2" layout="Heading 2">Ionization Energy</Text-field>We can compute the ionization energy by computing the energy of the ionized 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, the ionization energy is given 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which we can convert from hartrees to eVLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZDLUkjbWlHRiQ2JlEqYzJoMklFX2VWRicvJSVzaXplR1EjMTZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSJ+RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GOTYuUSomY29sb25lcTtGJ0YvRjxGPkZBRkNGRUZHRklGSy9GTlEsMC4yNzc3Nzc4ZW1GJy9GUUZWRjgtRiw2JlEoY29udmVydEYnRi9GMkY1LUkobWZlbmNlZEdGJDYlLUYjNigtRiw2JlEvYzJoMklFX2hhcnRyZWVGJ0YvRjJGNS1GOTYuUSIsRidGL0Y8Rj4vRkJGNEZDRkVGR0ZJRktGTS9GUVEsMC4zMzMzMzMzZW1GJy1GLDYmUSZ1bml0c0YnRi9GMkY1Rl1vLUYsNiZRI2VWRidGL0YyRjVGPEYvRjwtRjk2LlEiO0YnRi9GPEY+RmBvRkNGRUZHRklGS0ZNRlctSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGTy8lJmRlcHRoR0ZhcC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJ0Y4LUYsNiZRKmMyaDRJRV9lVkYnRi9GMkY1RjhGUkY4RlgtRmZuNiUtRiM2KC1GLDYmUS9jMmg0SUVfaGFydHJlZUYnRi9GMkY1Rl1vRmNvRl1vRmZvRjxGL0Y8RmlvRlxwRjgtRiw2JlEqYzJoNklFX2VWRidGL0YyRjVGOEZSRjhGWC1GZm42JS1GIzYoLUYsNiZRL2MyaDZJRV9oYXJ0cmVlRidGL0YyRjVGXW9GY29GXW9GZm9GPEYvRjxGaW8tRl1wNiZGX3BGYnBGZHAvRmdwUSVhdXRvRidGOEZccEZdci1GLDYjUSFGJ0YvLyUrZXhlY3V0YWJsZUdGQEY8Results
The relative ground state energy and ionization energy for three molecules: acetylene, methane, and ethane were calculated using DFT with seven different functionals: LDA,BP86, PBE, PBE0, B3LYP, B97-1, and B97-2. The functional BP86 and PBE are GGA functional, while PBE0, B3LYP, B97-2, and B97-1 are hybrid functional. The basis set used is 6-31g.
Figure 1 shows the comparison of relative ground state energy between the seven distinct functionals. It is observed that the energy of LDA is vastly different from the rest of the functionals. Figure 2 visualizes the effect of hybrid functionals on the ionization energy of the three molecules. It is found that for C2H2 and C2H4, functional B72 gives the lowest error, while B3LYP gives the highest error; however, for C2H6, B3LYP gives the lowest error, while PBE0 gives the highest 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.
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style="Heading 2" layout="Heading 2">Discussing conclusion
<Font size="12" bold="false">
</Font><Font size="14" bold="false">Inspired by Boese and et al. 2003 paper, this project explores the effect of different functional on
properties like ground state energy, and ionization energy</Font><Equation executable="true" style="Heading 2" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSIyRidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbW9HRiQ2L1EifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZLLUYjNictSSNtbkdGJDYmUSIyRidGMkY1RjhGMkY1LyUnaXRhbGljR1EldHJ1ZUYnL0Y5USdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjNRIzEyRidGNUY4</Equation><Font size="14" bold="false">. Boese have created a ranking for functionals for both GGA and hybrid functional. In this work, the general trend of accuracy of functionals is observed: where the accuracy of GGA and Hybrid functionals are way higher than LDA, and B-971 gives the most accurate result for acetylene and ethylene.
Admittedly, before conducting the project, I have already predicted that the values calculated by LDA would not give the best result, as the approximation of homogeneous electron gas does not capture the inhomogeneity in these molecules. Then why is the LDA functional still used? The reason is LDA is the only approximation that the exchange-correlation functional is known to a very high accuracy, as discussed in the method section. Recall that the challenge in DFT is that exact functional is unknown, and the explicit expressions for exchange-correlation functional are hard to obtain, thereby, even though LDA has limitations, it is still a good idea to test this functional. </Font></Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=<Text-field style="Heading 2" layout="Heading 2">Reference </Text-field>[1]Bickelhaupt, FM, 'A Chemist's Guide to Density Functional Theory, by Wolfram Koch and Max C. Holthausen [Review of: Wolfram Koch and Max C. Holthausen. A Chemist's Guide to Density Functional Theory]', Chemical Physics, vol. 2, pp. 257-258 (2001).
[2]Boese, A. D. et al. \342\200\234The role of the basis set: Assessing density functional theory.\342\200\235 Journal of Chemical Physics 119: 3005-3014 (2003).
[3]Chatfield, D. Christopher J. Cramer: Essentials of Computational Chemistry: Theories and Models . Theor Chem Acc 108, 367\342\200\223368 (2002).
[4]Frank Jensen. Introduction to Computational Chemistry. John Wiley & Sons, Inc., Hoboken, (NJ, USA, 2006).
[5]J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992).
[6]Wikipedia contributors, "Density functional theory," Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/w/index.php?title=Density_functional_theory&oldid=999780481 (accessed February 22, 2021).
[7]https://webbook.nist.gov/cgi/cbook.cgi?ID=C74862&Mask=20 (accessed February 22, 2021).
[8]https://webbook.nist.gov/cgi/cbook.cgi?ID=C74828&Mask=20 (accessed February 22, 2021).
[9]https://webbook.nist.gov/cgi/cbook.cgi?ID=C74840&Mask=20 (accessed February 22, 2021).LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=