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## Generalized Gradient Approximation Made Simple: The PBE Density Functional

Cabrini College, US has reference to this Academic Journal, Generalized Gradient Approximation Made Simple: The PBE Density Functional Rick Muller Quantum Chemistry Group Materials in addition to Process Simulation Center May 2000 Density Functional Theory H = T + VN + VJ + VX + VC T, VN, VJ are identical so that their HF counterparts VX is the Exchange Functional VC is the Correlation Functional Local Density Approximation Simplest Approximation Consider only the density at any point in space Generalized Gradient Approximation More Accurate Approximation Consider both the density in addition to the gradient of the density Terms in Density Functionals r Local density rs Seitz radius = (3/4pr)1/3 kF Fermi wave number = (3p2r)1/3 t Density gradient = |grad r|/2fksr z Spin polarization = (rup – rdown)/r f Spin scaling factor = [(1+z)2/3 + (1-z)2/3]/2 ks Thomas-Fermi screening wave number = (4kF/pa0)1/2 s Another density gradient = |grad r|/2kFr

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Local Density Approximation Local Spin Density Approximation Local Spin Density Correlation Functional Not in consideration of the faint of heart:

Generalized Gradient Approximation Functionals The GGA Correlation functional has the form The GGA Exchange functional has the form Limits the Correlation Functional Must Obey Slowly Varying Limit Rapidly Varying Limit Uniform Scaling so that High Density Limit Correlation: Slowly Varying Limit

Chapter 9 Plant Assets, Natural Resources, in addition to Intangible Assets Plant Assets, Natural Resources in addition to Intangible Assets Plant Assets, Natural Resources in addition to Intangible Assets (Contd.) Topics of Plant Assets (i.e., Property, Plant, in addition to Equipment or PPE) Topics of Intangible Assets A.1 PPE Valuation (APB Opinion No. 6) PPE Valuation (contd.) A.2 Determination of Acquisition Cost Capitalizing the Cost of Interest Example Example (contd.) B. Lump Sum Purchase B. Lump Sum Purchase (contd.) C. Depreciation of P.P.E & Depletion of Nature Resources C. Depreciation of P.P.E & Depletion of Nature Resources (contd.) C. Depreciation of P.P.E & Depletion of Nature Resources (contd.) C. Depreciation of P.P.E & Depletion of Nature Resources (contd.) Depreciation (For Financial Reporting Purposes) Example 1 Example 2 (partial year depreciation) Example 2 (contd.) – Presentation (Book Value (Carrying Value) = Cost – Acc. Depr) 1b. Sum-of-the-year?s-Digits Method (SYD) 1c. Declining-Balance Method Example Double Declining- Balance Method Example 1 Example 1 (contd.) A Comparison of Depreciation Methods Changes in Depreciation Estimate Example: (S-L Depr. Method) Example: (contd.) 2. Activity-Based Method: Example: Tax Depreciation MACRS MACRS (Continued) D. Disposal of Plant Assets D. Disposal of Plant Assets (contd.) Exchanging of Assets Exchanging of Assets (contd.) E. Capital Expenditures Versus Revenue Expenditures 1. Costs Subsequent so that Acquisition 2. Types of costs occurred subsequent so that acquisition: Accounting in consideration of Natural Resources in addition to Depletion Accounting in consideration of Intangible Assets Amortization of Intangibles 1. Patents: 2. Copyrights: 3. Trademarks & Trade Names: 4. Leaseholds 5. Franchises & Licenses: Franchises & Licenses (contd.): 6. Computer Software Costs: 7. Goodwill (not subject so that amortization ):

Correlation: Rapidly Varying Limit Correlation: Uniform Scaling so that High Density Limit PBE Correlation Correction Form

Limits the Exchange Functional Must Obey Uniform Scaling so that High Density Limit Spin Scaling Relationship Lieb-Oxford Bound Exchange: Uniform Scaling so that High Density Limit Exchange: Spin Scaling Relationship

Exchange: Small Density Variations Must Reproduce Uniform Density Limit Lieb-Oxford Bound PBE Exchange Correction Form

Comparison of Density Functionals References J. P. Perdew, K. Burke, M. Enzerhof. “Generalized Gradient Approximation Made Simple.” PRL 77, 3865 (1996) J. P. Perdew, in “Electronic Structure of Solids ’91.” P. Ziesche, H. Eschrig eds. (Akademie Verlag, Berlin, 1991), p. 11. B. G. Johnson, P. M. W. Gill, J. A. Pople. “The Performance of a Family of Density Functionals.” JCP 98, 5612 (1993). T. Ziegler. “Approximate Density Functional Theory as a Practical Tool in Molecular Energetics in addition to Dynamics.” Chem. Rev. 91, 651 (1991). T. V. Russo, R. L. Martin, P. J. Hay. “Density Functional Calculations on First-Row Transition Metals.” JCP 101, 7729 (1994).

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