1 Unform Electron Gas Ths chapter llustrates the dea that all propertes of the homogeneous electron gas (HEG) can be calculated from electron densty. Intutve Representaton of Densty Electron densty n s a very physcally meanngful quantty, but just to make t more ntutve (and even measurable!), let s defne such that r s ( 3 4πn )3 (1) 4 3 πr3 s = 1 n = Ω N e (2) Snce n s unform n HEG, r s s a vald alternatve representaton of densty. Electron Hamltonan Recall the electron Hamltonan (frst lne of (3.1) on p52) H e = 2 2 + 1 k e2 2m e 2 r j,i j k Z Ie 2 r I (3) n atomc unts H e = 1 2 + 1 1 2 2 r j,i j Z I r I (4) when both the electrons and the background ons are unformly dstrbuted, the last term (electron-on nteracton) can be approxmated by densty nteracton H e = 1 2 + 1 1 1 2 2 r j 2 j d rd r n 2 r r (5) ne ne Fgure 1: Unform electrons and ons
2 now, f we use r s as the length unt nstead of Bohr a o H e = 1 ( a o ) 2 2 2 r + 1 ( a o ) 1 1 s 2 r s r j 2 (a o ) d rd r n 2 r s r r j =( a o ) 2 1 2 + 1 r s 1 1 d rd r n 2 r s 2 2 a o r j 2 r r j =( a o ) 2 1 r s 2 2 + 1 r s 1 1 3 d r n2 (6) 2 a o r j 2 2π r j We see that as we ncrease densty (decrease r s ), the total energy of HEG goes up (duh), but ts behavor also changes due to a rescalng of the relatve strengths of the knetc and nteracton terms. Non-nteractng Lmt Wthout nteracton, snce the system s homogeneous each electron feels a background potental V o and the Hamltonan s smple H NI e = ˆT = 1 2 the egenfunctons are plane waves and the egenvalues are ψ k ( r) = 1 Ω e k r 2 + V o (7) (8) ɛ k = 1 2 k2 (9) For all Ne σ electrons wth the same spn σ, we can calculate the Ferm vector due to Paul excluson as 4π 3 kσ 3 F = (2π)3 Ω N e σ = (2π) 3 n σ kf σ = (6π 2 ) 1/3 n σ1/3 (10) snce each k state has sze (2π)3 Ω n recprocal space. The Ferm energy per electron EF σ o = Eσ F /N e σ s gven by EF σ o = 1 2 kσ 2 F (11) and knetc energy per electron T σ o = 1 N σ e ɛ k ( k k σ F 0 d k (2π) 3 /Ω ) 1 k σ F 0 d k 1 (2π) 3 /Ω 2 k2 =( 4π 3 k3 F ) 1 4π 5 (1 2 k5 F ) = 3 5 Eσ F o (12)
3 Hartree-Fock Approxmaton Wth the ncluson of coulomb and exchange nteractons, the Hamltonan H HF e = ˆT + ˆV ext + ˆV x (13) the egenfunctons are stll plane waves ψ k ( r) = 1 Ω e k r (14) but the egenvalues are modfed by matrx elements of the exchange operator ɛ k = 1 2 k2 + k F f(x) (15) π where x = k/k F and Luttnger guarantees that k F s stll (10), further f(x) = (1 + 1 x2 2x + x ln 1 ) (16) 1 x Notce lm f(x) = 2 and lm f(x) = 1, thus as k s vared from 0 to k F, x 0 x 1 ɛ k s lowest reach s extended by 2k F /π, whereas ts hghest reach s reduced by k F /π, resultng n a net broadenng of allowed energy band wdth W = k F /π. If we normalze (15) wth the Ferm energy ɛ F = 1 2 k2 F, we get η(x) = ɛ k /ɛ F = x 2 + 2 πk F f(x) (17) If we further measure energy from the Ferm surface ( η(1) = 0), we need η(x) = (x 2 1) + 2 πk F (f(x) + 1) (18) Ths Normalzed energy s plotted n the book (Fgure 5.2 Rght). Exchange Energy Recall the k F π f(x) contrbuton to energy comes from the exchange operator, therefore the exchange energy per electron ɛ σ x = 1 2 kσ F π f(x) = 3 4π kσ F = 3 4 ( 6 π nσ ) 1/3 (19) notce the factor 1/2 s to elmnate over-count and the average value of f(x) throughout the Ferm sphere f(x) = ( 4 kf 3 πk3 F ) 1 d kf(k) = 3 2 The average exchange energy per electron for both and electrons s ɛ x =E x /N = N ɛ x + N ɛ x N = 3 4 ( 6 π )1/3 n 4/3 + n 4/3 0 n = n ɛ x + n ɛ x n (20) (21)
4 If we defne the degree of polarzaton to be ζ = n n n, we can rewrte (21) as ɛ x = 3 4 ( 6 π )1/3 1 2 4/3 [(1 + ζ)4/3 + (1 ζ) 4/3 ]n 1/3 =An 1/3 1 2 [(1 + ζ)4/3 + (1 ζ) 4/3 ] (22) at the un-polarzed and fully-polarzed extremes { ɛx (n, 0) = An 1/3 ɛ x (n, 1) = An 1/3 2 1/3 (23) for some nexplcable reason, we want to rewrte (22) as an nterpolaton ɛ x (n, ζ) = ɛ x (n, 0) + [ɛ x (n, 1) ɛ x (n, 0)] { 1 2 (1 + ζ) 4/3 + (1 ζ) 4/3 } 2 2 1/3 1 (24) Exchange Hole The Ferm exchange hole can be calculated as (see (3.56) on p66) g( x; x ) = 1 ρ σ( x, x ) 2 n( x)n( x ) gx σ,σ (y) = 1 ( ρ(y) n )2 (25) where y = k σ F r wth r = r r beng the separaton between electrons and ρ(y) s the frst order reduced densty matrx for ndependent Fermons gven by ρ(r = r r ) = d k r = 1 (2π) 3 k = β 1 d 1 d (2π) 2 r dr r dr f(ɛ(k)) k r d kf(ɛ(k))e k ( r r ) dk cos(kr)f (β( 1 2 k2 µ)) (26) 1 where f(ɛ) = r e β(ɛ µ) +1 s the Ferm-Drac dstrbuton and 1 k = e k r (2π) 3 are just plane waves. The last lne was obtaned usng partal ntegraton and ths fnal form s REALLY revealng! ρ(y) = k F 3π y 3 2 [3sn(y) y cos(y) Fnally, pluggng (27) nto (25) we have ] = n 2 sn(y) y cos(y) [3 y 3 ] (27) g σ,σ x (y) = 1 [3 sn(y) y cos(y) y 3 ] 2 (28)
5 Correlaton Energy In general, t s mpossble to determne the correlaton energy analytcally. Exact solutons are only avalable n certan lmts. For un-polarzed gas ζ = 0, results are avalable for the hgh densty (r s 0) and low densty case (r s 1) { 0.311 ln(rs ) 0.048 + r s (A ln(r s ) + C) + r s 0 ɛ c (r s, 0) a 1 r s + a2 + a3 rs 3/2 r + r s 2 s 1 (29) Some data are also avalable for fully polarzed gas ɛ c (n, 1). Thus Perdew- Zunger(PZ) boldly (and qute correctly) guessed that ɛ c nterpolates n the same way that ɛ x does. They proposed ɛ c (n, ζ) = ɛ c (n, 0) + [ɛ c (n, 1) ɛ c (n, 0)]f x (ζ) (30) where f x (ζ) s the same nterpolaton functon at the end of (24). Correlaton Hole The total contrbuton from exchange and correlaton can be expressed as an ntegral of exchange-correlaton hole ɛ xc (r s ) = 1 2n e2 d r nav xc(r) (31) r where n av xc(r) s the couplng-constant-averaged hole Bondng n Metals n av xc(r) = 1 0 dλn λ xc(r) (32) The alkal metals are remarkably well represented by energy of HEG plus attractve nteracton wth the postve cores. Wth ths model, the total energy per electron ɛ tot = E total N = 1.105 rs 2 0.458 + ɛ c 1 r s 2 α + ɛ R (33) r s where ɛ c s the correlaton energy per electron, α s the Madelung constant for pont charges and ɛ R s due to core repulson owng to the fnte szes of ons, whch amounts to removng attracton n regons wth core radus R C around the ons Rc ɛ R = n2π drr 2 e2 0 r = 3 Rc 2 2 rs 3 (34) We can now predct the densty of these metals by fndng the mnmum of (33), takng α = 1.80 and R c = 2, we get r s = 4 whch s not bad. Further, the predcted bulk modulus B = Ω d2 E dω 2 = 51.7 Mbar (35) s often qute good for sp-bonded metals to strongly bonded covalent solds. r 5 s
6 Exctaton Here we re manly nterested n collectve exctatons that don t change the number of electrons. Examples are charge densty fluctuatons (plasma oscllaton) descrbed by delectrc functon and spn fluctuatons descrbed by spn response functons. In a homogeneous system, the delectrc functon s smply a response to an nternal feld. In Fourer space In terms of potentals D(q, ω) = E(q, ω) + 4π P (q, ω) = ɛ(q, ω) E(q, ω) (36) ɛ(q, ω) = δv ext( q, ω) δv test ( q, ω) = 1 4πe2 q 2 χ n( q, ω) (37)