Abstract
Density functional theory is based on the two Hohenberg–Kohn theorems, which state that the ground-state properties of an atom or molecule are determined by its electron density function, and that a trial electron density must give an energy greater than or equal to the true energy (the latter theorem is true only if the exact functional could be used). In the Kohn–Sham approach the energy of a system is formulated as a deviation from the energy of an idealized system with noninteracting electrons. From the energy equation, by minimizing the energy with respect to the Kohn–Sham orbitals the Kohn–Sham equations can be derived, analogously to the Hartree–Fock equations. Finding good functionals is the main problem in DFT. Various levels of DFT and new functionals are discussed. The mutually related concepts of electronic chemical potential, electronegativity, hardness, softness, and the Fukui function are exemplified.
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- 1.
Walter Kohn, born in Vienna 1923. B.A., B.Sc., University of Toronto, 1945, 1946. Ph.D. Harvard, 1948. Instructor in physics, Harvard, 1948–1950. Assistant, Associate, full Professor, Carnegie Mellon University, 1950–1960. Professor of physics, University of California at San Diego, 1960–1979; University of California at Santa Barbara 1979-present. Nobel Prize in chemistry 1998.
- 2.
M06:“M zero six”, or colloquially “M oh six”. A descendant of M05, Minnesota ‘05 (2005): Y. Zhao, N.E. Schultz, D.E. Truhlar, J. Chem. Phys., 2005, 123, 161103.
- 3.
Personal communication from Professor J.P. Perdew, 2009 November 7.
- 4.
Kenichi Fukui, born Nara, Japan, 1918. Ph.D. Kyoto Imperial University 1948, Professor Kyoto Imperial University 1951. Nobel Prize 1981. Died 1998.
References
(a) Whitaker A (1996) Einstein, Bohr and the quantum dilemma. Cambridge University Press, Cambridge (b) Yam P (June 1997) Scientific American, p 124 (c) Albert DZ (May 1994) Scientific American, p 58 (d) Albert DZ (1992) Quantum mechanics and experience. Harvard University Press, Cambridge, MA (e) Bohm D, Hiley HB (1992) The undivided universe. Routledge, New York (f) Baggott J (1992) The meaning of quantum theory. Oxford, New York (g) Jammer M (1974) The philosophy of quantum mechanics. Wiley, New York
Bader RFW (1990) Atoms in molecules. Oxford, New York
Reference 2, pp 7–8
Shusterman GP, Shusterman AJ (1997) J Chem Educ 74:771
Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford, New York, p 53
(a) Wilson E (1998) Chem Eng News October 19:12 (b) Malakoff D (1998) Science 282:610
See e.g. Diacu F (1996) Mathematical Intelligencer 18:66
Kohn W (1951) Phys Rev 84:495
Cf. Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, p 422, equation (13.130); cf. p 624, problem 15.67, for the Kohn–Sham orbitals. The multielectron wavefunction is treated on pp 421–423
Löwdin P-O (1955) Phys Rev 97:1474
Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford, New York
Koch W, Holthausen M (2000) A chemist’s guide to density functional theory. Wiley-VCH, New York
Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, pp 573–592
Friesner RA, Murphy RB, Beachy MD, Ringnalda MN, Pollard WT, Dunietz BD, Cao Y (1999) J Phys Chem A 103:1913
Kohn W, Becke AD, Parr RG (1996) J Phys Chem 100:12974
Parr RG, Yang W (1995) Annu Rev Phys Chem 46:701
Cramer CJ (2004) Essentials of computational chemistry, 2nd edn. Wiley, New York, Chapter 8
Jensen F (2007) Introduction to computational chemistry, 2nd edn. Wiley, New York, Chapter 6
E.g. Griffiths DJ (1995) Introduction to quantum mechanics. Prentice-Hall, Engelwood Cliffs, NJ
Earlier work (1927) by Fermi was published in Italian and came to the attention of the physics community with a paper in German: Fermi E (1928) Z Phys 48:73. This appears in English translation in March NH (1975) Self-consistent fields in atoms. Pergamon, Oxford
Thomas LH (1927) Proc Camb Phil Soc 23:542; reprinted in March NH (1975) Self-consistent fields in atoms. Pergamon, Oxford
Reference 11, Chapter 6
Slater JC (1975) Int J Quantum Chem Symp 9:7
Reviews: (a) Connolly JWD (1977). In: Segal GA (ed) Semiempirical methods of electronic structure calculations part A: techniques. Plenum, New York. (b) Johnson KH (1973) Adv Quantum Chem 7:143
Slater JC (1951) Phys Rev 81:385
For a personal history of much of the development of quantum mechanics, with significant emphasis on the Xα method, see: Slater JC (1975) Solid-state and molecular theory: a scientific biography. Wiley, New York
Reference 13, pp 573–577
Hohenberg P, Kohn W (1964) Phys Rev B 136:864
Reference 11, section 3.4
Kohn W, Sham LJ (1965) Phys Rev A 140:1133
Reference 13, section 13.14
Reference 18, p 241
Reference 11, sections 7.1–7.3
Reference 13, section 11.8
Reference 11, chapter 7
Reference 11, Appendix A
Reference 13, p 580
Merrill GN, Gronert S, Kass SR (1997) J Phys Chem A 101:208
Reference 11, p 185
Genesis 28. 10–12
The concept was apparently first enunciated by J. P. Perdew at the DFT2000 symposium in Menton, France. It first appeared in print in: Perdew JP, Schmidt K (2001). In: Van Doren VE, Van Alsenoy K, Geerlings P (eds) Density functional theory and its applications to materials. AIP Press, New York
Mattsson AE (2002) Science 298:759
Jensen F (2007) Introduction to computational chemistry, 2nd edn. Wiley, New York, pp 244–245
Sousa SF, Fernandes OA, Ramos MJ (2007) J Phys Chem A 111:10439
Zhao Y, Truhlar DG (2007) Acc Chem Res 41:157
Riley KE, Op’t Holt BT, Merz KM Jr (2007) J Chem Theory Comput 3:404
Perdew JP, Ruzsinszky A, Tao J, Staroverov VN, Scuseria GE, Csonka GI (2005) J Chem Phys 123:062201
Kurth S, Perdew JP, Blaha P (1999) Int J Quant Chem 75:889
Taylor AE (1955) Advanced calculus. Blaidsell Publishing Company, New York, p 371
Reference 11, pp 173–174
Vosko SH, Wilk L, Nusair M (1980) Can J Phys 58:1200
St-Amant A (1996). In: Lipkowitz KB, Boyd DB (eds) Reviews in computational chemistry, vol 7, Chapter 5. VCH, New York, p 223
Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ, pp 587
Becke AD (1988) Phys Rev A 38:3098
Head-Gordon M (1996) J Phys Chem 100:13213
Brack M, Jennings BK, Chu YH (1976) Phys Lett 65B:1
Becke AD (1993) J Chem Phys 98:1372, 5648
Stephens PJ, Devlin JJ, Chabalowski CF, Frisch MJ (1994) J Phys Chem 98:11623
E.g. Perdew JP (1995) Nonlocal density functionals for exchange and correlation: theory and application. In: Ellis DE (ed) Density functional theory of molecules, clusters, and solids. Kluwer, Dordrecht, The Netherlands
Schwabe T, Grimme S (2007) Phys Chem Chem Phys 9:3397, and references therein. B2PLYP does have empirical parameters, albeit just two
Wennmohs F, Neese F (2008) Chem Phys 343:217
Clark T (2000) J Mol Struct (Theochem) 530:1
Nooijen M (2009) Adv Quant Chem 56:181
Dewar MJS (1992). In: Seeman JI (ed) “A semiempirical life”, profiles, pathways and dreams series. American Chemical Society, Washington, DC, p 185
Zhao Y, Truhlar DG (2008) Theor Chem Account 120:215
Levine IN (2000) Quantum chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ; cited in index
Hehre WJ (1995) Practical strategies for electronic structure calculations. Wavefunction Inc, Irvine, CA
Hehre WJ, Lou L (1997) A guide to density functional calculations in Spartan. Wavefunction Inc, Irvine CA
Hehre WJ, Radom L, Schleyer pvR, Pople JA (1986) Ab initio molecular orbital theory. Wiley, New York; section 6.2
H 2 C=CHOH reaction: The only quantitative information on the barrier for this reaction seems to be: Saito S (1976) Chem Phys Lett 42:399, halflife in the gas phase in a Pyrex flask at room temperature ca. 30 minutes. From this one calculates (Chapter 5, Section 5.5.2.2d, Eq. 5.202) a free energy of activation of 93 kJ mol−1. Since isomerization may be catalyzed by the walls of the flask, the purely concerted reaction may have a much higher barrier. This paper also shows by microwave spectroscopy that ethenol has the O–H bond syn to the C=C. The most reliable measurement of the ethenol/ethanal equilibrium constant, by flash photolysis, is 5.89 × 10−7 in water at room temperature (Chiang Y, Hojatti M, Keeffe JR, Kresge AK, Schepp NP, Wirz J (1987) J Am Chem Soc 109:4000). This gives a free energy of equilibrium of 36 kJ mol−1 (ethanal 36 kJ mol−1 below ethenol). The accurate G3MP2 method [Chapter 5, Section 5.5.2.2b] places the gas phase free energy of ethanal 43 kJ mol−1 below that of ethenol. HNC reaction: The barrier for rearrangement of HNC to HCN has apparently never been actually measured. The equilibrium constant in the gas phase at room temperature was calculated (Maki AG, Sams RL (1981) J Chem Phys 75:4178) at 3.7 × 10−8, from actual measurements at higher temperatures; this gives a free energy of equilibrium of 42 kJ mol−1 (HCN 42 kJ mol−1 below HNC). The G3MP2 method places the gas phase free energy of HCN 59 kJ mol−1 below that of HNC. CH 3 NC reaction: The reported experimental activation energy is 161 kJ mol−1 (Wang D, Qian X, Peng J (1996) Chem Phys Lett 258:149; Bowman JM, Gazy B, Bentley JA, Lee TJ, Dateo CE (1993) J Chem Phys 99:308; Rabinovitch BS, Gilderson PW (1965) J Am Chem Soc 87:158; Schneider FW, Rabinovitch BS (1962) J Am Chem Soc 84:4215). The energy difference between CH3NC and CH3CN has apparently never been actually measured. The G3MP2 method places the gas phase free energy of CH3CN 99 kJ mol−1 below that of CH3NC. Cyclopropylidene reaction: Neither the barrier nor the equilibrium constant for the cyclopropylidene/allene reaction have been measured. The only direct experimental information of these species come from the failure to observe cyclopropylidene at 77 K (Chapman OL (1974) Pure Appl Chem 40:511). This and other experiments (references in Bettinger HF, Schleyer PvR, Schreiner PR, Schaefer HF (1997) J Org Chem 62:9267 and in Bettinger HF, Schreiner PR, Schleyer PvR, Schaefer HF (1996) J Phys Chem 100:16147) show that the carbene is much higher in energy than allene and rearranges very rapidly to the latter. Bettinger et al. 1997 (above) calculate the barrier to be 21 kJ mol−1 (5 kcal mol−1). The G3MP2 method places the gas phase free energy of allene 283 kJ mol−1 below that of cyclopropylidene
Spartan is an integrated molecular mechanics, ab initio and semiempirical program with an outstanding input/output graphical interface that is available in UNIX workstation and PC versions: Wavefunction Inc., http://www.wavefun.com, 18401 Von Karman, Suite 370, Irvine CA 92715, USA
Perdew JP, Burke K, Ernzerhof M (1996) Phys Rev Lett 77:3865; Erratum: Perdew JP, Burke K, Ernzerhof M (1997) Phys Rev Lett 78;1396
Tao J, Pewdew JP, Staroverov VN, Scuseria GE (2003) Phys Rev Lett 91:146401
Scheiner AC, Baker J, Andzelm JW (1997) J Comput Chem 18:775
El-Azhary AA (1996) J Phys Chem 100:15056
Bauschlicher CW Jr, Ricca A, Partridge H, Langhoff SR (1997). In: Chong DP (ed) Recent advances in density functional methods. Part II. World Scientific, Singapore
Scott AP, Radom L (1996) J Phys Chem 100:16502
As of mid-2009, the latest “full” version (as distinct from more frequent revisions) of the Gaussian suite of programs was Gaussian 09. Gaussian is available for several operating systems; see Gaussian, Inc., http://www.gaussian.com, 340 Quinnipiac St., Bldg. 40, Wallingford, CT 06492, USA
Ochterski JW, Gaussian white paper “Thermochemistry in Gaussian”, http://www.gaussian.com/g_whitepap/thermo.htm
Blanksby SJ, Ellison GB (2003) Acc Chem Res 36:255; Chart 1
Hammond GS (1955) J Am Chem Soc 77:334
From the NIST website, http://webbook.nist.gov/chemistry/: Chase MW Jr (1998) NIST-JANAF Themochemical tables, 4th edn. J Phys Chem Ref Data, Monograph 9, 1–1951
Peterson GA (1998). Irikura KK, Frurip DJ (eds) Computational thermochemistry, Chapter 13. American Chemical Society, Washington, DC
Goldstein E, Beno B, Houk KN (1996) J Am Chem Soc 118:6036
Martell JM, Goddard JD, Eriksson L (1997) J Phys Chem 101:1927
The data are from Hehre WJ (1995) Practical strategies for electronic structure calculations. Wavefunction, Inc., Irvine, CA; Chapter 4. In each case, the first 10 examples from the relevant table were used
Wiberg KB, Ochterski JW (1997) J Comp Chem 18:108
Rousseau E, Mathieu D (2000) J Comp Chem 21:367
Ventura ON, Kieninger M, Cachau RE (1999) J Phys Chem A 103:147
For this and other misgivings about the multistep methods see Cramer CJ (2004) Essentials of computational chemistry, 2nd edn. Wiley, Chichester, UK, pp 241–244
CBS-QB3 was found to give unacceptable errors for halogenated compounds: Bond D (2007) J Org Chem 72:7313
For pericyclic reactions: Ess DH, Houk KN (2005) J Phys Chem A 109:9542
Montgomery JA Jr, Frisch MJ, Ochterski JW, Petersson GA (1999) J Chem Phys 110:2822
del Rio A, Bourcekkine A, Meinel J (2003) J Comp Chem 24:2093
Singleton DA, Merrigan SR, Liu J, Houk KN (1997) J Am Chem Soc 119:3385
Glukhovtsev MN, Bach RD, Pross A, Radom L (1996) Chem Phys Lett 260:558
Bell RL, Tavaeras DL, Truong TN, Simons J (1997) Int J Quantum Chem 63:861
Truong TN, Duncan WT, Bell RL (1996). In: Laird BB, Ross RB, Ziegler T (eds) Chemical applications of density functional theory. American Chemical Society, Washington, DC
Zhang Q, Bell RL (1995) J Phys Chem 99:592
Eckert F, Rauhut G (1998) J Am Chem Soc 120:13478
Baker J, Muir M, Andzelm J (1995) J Chem Phys 102:2063
Jursic BS (1996). In: Seminario JM (ed) Recent developments and applications of modern density functional theory. Elsevier, Amsterdam
Brown SW, Rienstra-Kiracofe JC, Schaefer HF (1999) J Phys Chem A 103:4065
Cramer CJ (2004) Essentials of computational chemistry, 2nd edn. Wiley, Chichester, UK, p 309
Geerlings P, De Profit F, Martin JML (1996). In Seminario JM (ed) Recent developments and applications of modern density functional theory. Elsevier, Amsterdam
Lendvay G (1994) J Phys Chem 98:6098
Boyd RJ, Wang J, Eriksson LA (1995). In Chong DP (ed) Recent advances in density functional methods. Part I. World Scientific, Singapore
Levine IN (2000) Quantum Chemistry, 5th edn. Prentice Hall, Upper Saddle River, NJ; sections 9.9, 9.10
Stratman RE, Scuseria GE, Frisch MJ (1998) J Chem Phys 109:8218
Wiberg KB, Stratman RE, Frisch MJ (1998) Chem Phys Lett 297:60
Foresman JB, Frisch Æ (1996) Exploring chemistry with electronic structure methods. Gaussian Inc, Pittsburgh, PA, p 218
Jacquemin D, Preat J, Wathelet V, Fontaine M, Perpète EA (2006) J Am Chem Soc 128:2072
Zhao Y, Truhlar DG (2006) J Phys Chem A 110:13126
Cheeseman JR, Trucks GW, Keith TA, Frisch MJ (1996) J Chem Phys 104:5497
Frisch MJ, Trucks GW, Cheeseman JR (1996). In: Seminario JM (ed) Recent developments and applications of modern density functional theory. Elsevier, Amsterdam
Rablen PR, Pearlman SA, Finkbiner J (2000) J Phys Chem A 103:7357
Sefzik TH, Tureo D, Iuliucci RJ (2005) J Phys Chem A 109:1180
Wu A, Zhang Y, Xu X, Yan Y (2007) J Comp Chem 28:2431
Zhao Y, Truhlar D (2008) J Phys Chem A 112:6794
Pérez M, Peakman TM, Alex A, Higginson PD, Mitchell JC, Snowden MJ, Morao I (2006) J Org Chem 71:3103
Castro C, Karney WL, Vu CMH, Burkhardt SE, Valencia MA (2005) J Org Chem 70:3602
Silverstein RM, Bassler GC, Morrill TC (1981) Spectrometric Identification of organic compounds, 4th edn. Wiley, New York; methane, 191, 219; cyclopropane, 193, 220; benzene, 196, 222; acetone, 227
Patchkovskii S, Thiel W (1999) J Comp Chem 20:1220
Muchall HM, Werstiuk NH, Choudhury B (1998) Can J Chem 76:227
Levin RD, Lias SG (1982) Ionization potential and appearance potential measurements, 1971–1981. National Bureau of Standards, Washington, DC
Curtis LA, Nobes RH, Pople JA, Radom I (1992) J Chem Phys 97:6766
Golas E, Lewars E, Liebman J (2009) J Phys Chem A 113:9485
(a) Baerends EJ, Gritsenko OV (1997) J Phys Chem A 101:5383 (b) Chong DP, Gritsenko OV, Baerends EJ (2002) J Chem Phys 116:1760
Cramer CJ (2004) Essentials of computational chemistry, 2nd edn. Wiley, Chichester, UK, p 272
Stowasser R, Hoffmann R (1999) J Am Chem Soc 121:3414
Salzner U, Lagowski JB, Pickup PG, Poirier RA (1997) J Comp Chem 18:1943
Vargas R, Garza J, Cedillo A (2005) J Phys Chem A 109:8880
Zhan C-C, Nichols JA, Dixon DA (2003) J Phys Chem A 107:4184
Zhang GZ, Musgrave CB (2005) J Phys Chem A 111:1554
Hunt WJ, Goddard WA (1969) Chem Phys Lett 3:414
Berzelius JJ (1819) Essai sur la théorie des proportions chimiques et sur l’influence chimique de l’électricité; see Nye MJ (1993) From chemical philosophy to theoretical chemistry. University of California Press, Berkeley, CA, p 64
Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford, New York, pp 90–95
Mulliken RS (1952) J Am Chem Soc 74:811
(a) Pearson RG (1963) J Am Chem Soc 85:3533 (b) Pearson RG (1963) Science 151:172
Footnote in [139a], p 3533
(a) Pearson RG (1973) Hard and soft acids and bases. Dowden, Hutchinson and Ross, Stroudenburg, PA (b) Lo TL (1977) Hard and soft acids and basis in organic chemistry. Academic Press, New York
Dewar MJS (1992) A semiempirical life. American Chemical Society, Washington, DC, p 160
Ritter S (2003) Chem Eng News 17 Feb:50
Parr RG, Yang W (1984) J Am Chem Soc 106:4049
Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford, New York; Chapters 4 and 5 in particular
(a) Gibbs JW (1993) The scientific papers of J. Willard Gibbs: Vol. I, Thermodynamics. Ox Bow, Woodbridge CT. (b) Kaplan TA (2006) Confronting confusion about chemical potential. J Stat Phys 122:1237. (c) Job G, Herrmann F (2006) An attempt to give an intuitive feeling for chemical potential. Eur J Phys 27:353. (d) Baierlein R (2001) An explanation of chemical potential in different ways. Am J Phys 69:423
Iczkowski RP, Margrave JL (1961) J Am Chem Soc 83:3547
Parr RG, Donnelly RA, Levy M, Palke WE (1978) J Chem Phys 68:3801
Mulliken RS (1934) J Chem Phys 2:782
Pearson RG (1999) J Chem Educ 76:267
Parr RG, Pearson RG (1983) J Am Chem Soc 105:7512
Toro-Labbé A (1999) J Phys Chem A 103:4398
Nguyen LT, Le TN, De Proft F, Chandra AK, Langenaeker W, Nguyen MT, Geerlings P (1999) J Am Chem Soc 121:5992
(a) Fukui K (1987) Science 218:747. (b) Fleming I (1976) Frontier orbitals and organic chemical reactions. Wiley, New York. (c) Fukui K (1971) Acc Chem Res 57:4
Yang W, Mortier WJ (1986) J Am Chem Soc 108:5708
Carey FA, Sundberg RL (2000) Advanced organic chemistry, 3rd edn. Plenum, New York, p 437
Méndez F, Gázquez JL (1994) J Am Chem Soc 116:9298
Damoun S, Van de Woude G, Choho K, Geerlings P (1999) J Phys Chem A 103:7861
Anderson JSM, Melin J, Ayers PW (2007) J Chem Theory Comput 3:375
(a) Zhou Z, Parr RG (1989) J Am Chem Soc 111:7371 (b) Zhou Z, Parr RG, Garst JF (1988) Tetrahedron Lett 29:4843
Parr RG, Yang W (1989) Density-functional theory of atoms and molecules. Oxford, New York, p 101
Melin J, Ayers PW, Ortiz JV (2007) J Phys Chem A 111:10017
Padmanabhan J, Parthasarthi R, Elango M, Subramanian V, Krishnamoorthy BS, Gutierrez-Oliva S, Toro-Labbé A, Roy DR, Chattaraj PK (2007) J Phys Chem A 111:9130
Bulat FA, Chamorro E, Fuentealba P, Toro-Labbé A (2004) J Phys Chem A 108:342
Melin J, Aparicio F, Subramanian V, Galván M, Chattaraj PK (2004) J Phys Chem A 108:2487
(a) Koch W, Holthausen M (2000) A chemist’s guide to density functional theory. Wiley-VCH, New York, part B, and refs. therein (b) Frenking G (1997) J Chem Soc Dalton Trans 1653
(a) Kyistyan S, Pulay P (1994) Chem Phys Lett 229:175 (b) Perez-Jorda JM, Becke AD (1995) Chem Phys Lett 233:134
(a) Lozynski M, Rusinska-Roszak D, Mack H-G (1998) J Phys Chem A 102:2899 (b) Adamo C, Barone V (1997). In: Chong DP (ed) Recent advances in density functional methods. Part II. World Scientific, Singapore (c) Sim F, St-Amant A, Papai I, Salahub DR (1992) J Am Chem Soc 114:4391
Added in Press
Book: Chattaraj PK (ed) (2009) Chemical reactivity theory. CRC press, Boca Raton, FL. Historical account by Parr, then theoretical development of electron density properties and reactivity indexes. Review of book: Hackett JC (2010) J Am Chem Soc 132:7558
Book: Sholl D, Steckel JA (2009) Density functional theory. A practical introduction. Wiley, Hoboken. Useful introduction with practical examples. Review of book: Jungwirth P (2010) Angew Chem Int Ed Engl 49:485
Theophilou AK (2010) DFT without functional derivatives, a suggested approach. J Mol Struct (Theochem) 943:42
Li Y, Lu D, Nguye H-V, Galli G (2010) DFT and weak interactions, the EXX/RPA approach. J Phys Chem A 114:1944
Bader RFW (2010) Historical and conceptual review of the electron density function. J Mol Struct (Theochem) 943:2
Fuentealba P, David J, Guerra D (2010) Are DFT parameters thermodynamic or kinetic? Conclusion: “A combination of both depending on the type of reaction”. J Mol Struct (Theochem) 943:127
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Appendices
Easier Questions
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1.
State the arguments for and against regarding DFT as being more a semiempirical than an ab initio-like theory.
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2.
What is the essential difference between wavefunction theory and DFT? What is it that, in principle anyway, makes DFT simpler than wavefunction theory?
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3.
Why can’t current DFT calculations be improved in a stepwise, systematic way, as can ab initio calculations?
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4.
Which of these prescriptions for dealing with a function are functionals: (1) square root of f(x). (2) sinf(x). (3) \( \sum\limits_{x = 1}^3 {f(x)} \). (4) \( \int {f(x)dx} \). (5) exp(f(x)).
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5.
For which class(es) of functions is the nth derivative of f(x) a functional?
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6.
Explain why a kind of molecular orbital is found in current DFT, although DFT is touted as an alternative to wavefunction theory.
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7.
What is fundamentally wrong with functionals that are not gradient-corrected?
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8.
The ionization energy of a molecule can be regarded as the energy required to remove an electron from its HOMO. How then would a pure density functional theory, with no orbitals, be able to calculate ionization energy?
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9.
Label these statements true or false: (1) For each molecular wavefunction there is an electron density function. (2) Since the electron density function has only x, y, z as its variables, DFT necessarily ignores spin. (3) DFT is good for transition metal compounds because it has been specifically parameterized to handle them. (4) In the limit of a sufficiently big basis set, a DFT calculation represents an exact solution of the Schrödinger equation. (5) The use of very big basis sets is essential with DFT. (6) A major problem in density functional theory is the prescription for going from the molecular electron density function to the energy.
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10.
Explain in words the meaning of the terms electronegativity, hardness, and the Fukui function.
Harder Questions
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1.
It is sometimes said that electron density is physically more real than a wavefunction. Do you agree? Is something that is more easily grasped intuitively necessarily more real?
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2.
A functional is a function of a function. Explore the concept of a function of a functional.
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3.
Why is it that the Hartree–Fock Slater determinant is an inexact representation of the wavefunction, but the DFT determinant for a system of noninteracting electrons is exact for this particular wavefunction?
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4.
Why do we expect the “unknown” term in the energy equation (E XC[ρ 0 ], in Eq. 7.21) to be small?
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5.
Merrill et al. have said that “while solutions to the [HF equations] may be viewed as exact solutions to an approximate description, the [KS equations] are approximations to an exact description!” Explain.
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6.
Electronegativity is the ability of an atom or molecule to attract electrons. Why then is it (from one definition) the average of the ionization energy and the electron affinity (Eq. 7.32), rather than simply the electron affinity?
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7.
Given the wavefunction of a molecule, it is possible to calculate the electron density function. Is it possible in principle to go in the other direction? Why or why not?
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8.
The multielectron wavefunction Ψ is a function of the spatial and spin coordinates of all the electrons. Physicists say that Ψ for any system tells us all that can be known about the system. Do you think the electron density function ρ tells us everything that can be known about a system? Why or why not?
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9.
If the electron density function concept is mathematically and conceptually simpler than the wavefunction concept, why did DFT come later than wavefunction theory?
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10.
Is a metal, with its common pool of electrons, a good approximation of the homogeneous electron gas of early DFT theory? Why or why not?
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Lewars, E.G. (2011). Density Functional Calculations. In: Computational Chemistry. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3862-3_7
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