Chemistry 460 Fall 2017 Dr. Jea M. Stadard Nvember 29, 2017 A Hartree-Fck Calculati f the Water Mlecule Itrducti A example Hartree-Fck calculati f the water mlecule will be preseted. I this case, the water mlecule will have its gemetry fixed at the experimetal values f bd legths (R(O-H)=0.95 Å) ad bd agle ( H-O-H= 104.5 ). Thus, the electric eergy ad wavefucti will be cmputed fr fixed uclear psitis; this is kw as a sigle-pit eergy calculati. A miimal basis set f atmic rbital fuctis will be emplyed. Mlecular structure ad crdiates Fr the purpses f the example calculati, sftware packages such as Gaussia 09 psiti the water mlecule such that the atms lie i the yz-plae with the ceter f mass at the rigi; this is kw as the stadard rietati. I the case f water with the specified bd legths ad agles, the cartesia crdiates f the atms are shw i Table 1. Ntice that the y- ad z-crdiate values f the hydrge atms are symmetric r atisymmetric abut the xyge atm psiti, which allws fr easy iclusi f the C 2v symmetry aspects f the water mlecule i the Hartree-Fck calculatis. Table 1. Atmic crdiates (i Å) f the water mlecule i its stadard rietati (frm Gaussia 09 lg file). --------------------------------------------------------------------- Ceter Atmic Atmic Crdiates (Agstrms) Number Number Type X Y Z --------------------------------------------------------------------- 1 1 0 0.000000 0.751155-0.465285 2 8 0 0.000000 0.000000 0.116321 3 1 0 0.000000-0.751155-0.465285 --------------------------------------------------------------------- Atmic rbital basis fuctis The water mlecule has a ttal f 10 electrs, eight frm the xyge atm ad e each frm the hydrge atms. Therefre, fr a clsed shell mlecular system like water i its grud state with 10 ttal electrs, the wavefucti i the frm f a Slater Determiat is Ψ H2 O = 1 10! φ 1 φ 1 φ 2 φ 2 φ 3 φ 3 φ 4 φ 4 φ 5 φ 5. (1) The fuctis φ i fr water are mlecular rbitals defied usig the LCAO-MO apprximati, ( ) = c µi f µ ( 1) φ i 1 K. (2) µ=1 Here, the terms c µi crrespd t liear cefficiets, the fuctis f µ are the atmic rbital basis fuctis, ad K is the ttal umber f atmic rbital basis fuctis used t represet the mlecular rbitals. I this example calculati, a miimal basis set will be used which csists f 1s basis fuctis fr each H atm, ad the 1s, 2s, ad set f 2p (2p x, 2p y, 2p z ) basis fuctis fr O atm. I Gaussia 09, a typical basis set f this type is called STO-3G, ad fr water csists f 7 basis fuctis. The umberig f the basis fuctis fr the rest f this example is give i Table 2.
2 Table 2. Basis fuctis fr the HF/STO-3G calculati f the water mlecule. Basis fucti # Basis fucti type 1 1s O 2 2s O 3 2p x O 4 2p y O 5 2p z O 6 1s H a 7 1s H b Hartree-Fck-Rtha equatis Miimizig the expectati value fr the eergy f the Slater Determiat with the LCAO-MO apprximati fr the mlecular rbitals yields the Hartree-Fck-Rtha (H-F-R) equatis. Fr water with K=7 basis fuctis, the H-F-R equatis are 7 ( F µν ε i S µν ) c νi = 0, µ =1, 2, 7. (3) ν=1 Here, F µν are Fck itegrals, S µν are verlap itegrals, ε i are the rbital eergies, ad the c νi are the liear cefficiets. Overlap itegrals Defiig the terms i the H-F-R equatis, the verlap itegrals S µν are itegrals ver pairs f the atmic rbital basis fuctis, S µν = f µ (1) f ν (1). (4) Fr water, there are 7 7 = 49 verlap itegrals. Hwever, because the rder f the prduct f fuctis i the itegrad des t matter, we have that S µν = S νµ, ad therefre there are fewer uique values (28 ttal). The values f the verlap itegrals S µν ca be displayed i matrix frm where the first idex f the elemet ( µ i this case) crrespds t the rw ad the secd idex (ν i this case) crrespds t the clum i the matrix. Fr the STO-3G basis set with the basis fuctis specified i the rder give i Table 2, the verlap matrix S is shw i Figure 1. Nte that ly the lwer prti is shw because the upper prti is related by symmetry sice S µν = S νµ. S = # 1.000 0.237 1.000 0.000 0.000 1.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 1.000 0.055 0.479 0.000 0.313 0.242 1.000 0.055 0.479 0.000 0.313 0.242 0.256 1.000 & Figure 1. Overlap matrix S fr HF/STO-3G calculati f water.
I the verlap matrix, te that all the diagal elemets S µµ equal 1 because f rmalizati f the basis fuctis. I additi, each 2p-type basis fucti the xyge atm is rthgal ( S µν = 0 ) t the ther 2p-type basis fuctis ad t the 1s ad 2s basis fuctis the xyge atm. Nte hwever, that the 1s ad 2s basis fuctis xyge are t rthgal; their verlap crrespds t elemet S 21, which equals 0.237. This lack f rthgality is a result f the chice f a gaussia frm fr these basis fuctis istead f usig the eigefuctis f a e-electr Hamiltia peratr (which wuld be rthgal). This is cmm practice fr s-type basis fuctis the same uclear ceter, as well as whe multiple sets f p-type r higher agular mmetum basis fuctis are emplyed. Fially, it shuld als be ted that basis fuctis differet uclear ceters are t i geeral rthgal; thus, mst f the xyge atm basis fuctis verlap with thse the hydrge atms. The excepti is the xyge 2p x basis fucti, which gives zer verlap with the hydrge basis fuctis due t cacellati (equal but ppsite verlap abve ad belw the plae f the mlecule). 3 Fck itegrals The Fck itegrals F µν i Equati (3) are defied as F µν = H µν K K + P λσ [ µν λσ ( µλ νσ ) ]. (5) λ=1 σ =1 ( ) 1 2 The terms H µν crrespd t the e-electr Hamiltia itegrals, H µν = f µ ( 1) 1 2 M ˆ 2 1 Z α f ν 1 r α=1 α1 ( ). (6) Here, the term 1 2 ˆ 1 2 crrespds t the kietic eergy peratr fr electr 1, while the term Z α crrespds t the ptetial eergy peratr fr the electr-uclear attractis, where Z α is the atmic umber f ucleus α ad r α1 is the distace betwee ucleus α ad electr 1. The e-electr itergrals are usually evaluated i tw parts, the kietic eergy itegrals T µν ad the ptetial eergy itegrals V µν. M α=1 r α1 Kietic eergy itegrals As might be expected, the kietic eergy itegrals T µν are defied as T µν = f µ ( 1) 1 2 ˆ 1 2 f ν ( 1). (7) Fr a calculati f water with the STO-3G basis set as specified i Table 2, the kietic eergy matrix T is shw i Figure 2. Agai, ly the lwer prti is shw because the upper prti is related by symmetry sice T µν = T νµ. T = # 29.003 0.168 0.808 0.000 0.000 2.529 0.000 0.000 0.000 2.529 0.000 0.000 0.000 0.000 2.529 0.002 0.132 0.000 0.229 0.177 0.760 0.002 0.132 0.000 0.229 0.177 0.009 0.760 & Figure 2. Kietic eergy matrix T fr HF/STO-3G calculati f water.
4 Fr the kietic eergy elemets T µν, te that the diagal elemets are i geeral much larger i magitude tha the ff-diagal elemets, ad the diagal elemets are always psitive. The ff-diagal elemets are geerally small i magitude ad may be either psitive r egative. Ptetial eergy itegrals The ptetial eergy itegrals V µν are defied as V µν = f µ ( 1) M Z α f ν 1 r α1 α=1 ( ). (8) Fr ur water calculati at the HF/STO-3G level, the kietic eergy matrix T is shw i Figure 3. Agai, ly the lwer prti is shw because the upper prti is related by symmetry sice V µν = V νµ. V = # 61.733 7.447 10.151 0.000 0.000 9.926 0.000 0.000 0.000 10.152 0.019 0.226 0.000 0.000 10.088 1.778 3.920 0.000 0.228 0.184 5.867 1.778 3.920 0.000 0.228 0.184 1.652 5.867 & Figure 3. Ptetial eergy matrix V fr HF/STO-3G calculati f water. Here we see that the ptetial eergy diagal elemets V µµ are i geeral much larger i magitude tha the ffdiagal elemets. The diagal elemets are always egative; this is because the electr-uclear iteracti fr electrs i the same rbital is always attractive (i.e., egative). The ff-diagal elemets are agai small i magitude ad may be either psitive r egative. Oe-electr Hamiltia itegrals The e-electr kietic eergy ad ptetial eergy itegrals may be cmbied t frm the e-electr Hamiltia itegrals, H µν, usig the relati H µν = T µν + V µν, (9) r H = T + V i matrix frm. The e-electr itegrals H µν are shw i matrix frmat i Figure 4. H = # 32.730 7.615 9.343 0.000 0.000 7.397 0.000 0.000 0.000 7.623 0.019 0.226 0.000 0.000 7.559 1.780 3.788 0.000 0.001 0.007 5.107 1.780 3.788 0.000 0.001 0.007 1.643 5.107 & Figure 4. Oe-electr Hamiltia matrix H fr HF/STO-3G calculati f water.
Tw-electr itegrals The ext step is t cmpute the tw-electr itegrals frm Equati (5). The terms µν λσ ( ) ad µλ νσ ( ) represet tw-electr repulsi itegrals frm the Culmb ad Exchage terms i the Fck peratr, 5 ( µν λσ ) = f µ ( 1) f λ ( 2) ( µλ νσ ) = f µ ( 1) f ν ( 2) 1 r 12 f ν 1 ( ) f σ 2 ( ) 1 r 12 f λ 1 ( ) f σ 2 ( ). (10) The umber f tw-electr itegrals that must be cmputed is K 4, where K is the umber f basis fuctis. Fr the HF/STO-3G calculati f water, K=7, s the umber f tw-electr itegrals t be cmputed is 2401. Because f the symmetry f the water mlecule, this umber is reduced t a mere 406 itegrals. Eve that may wuld take a lt f space t list a page, s their umerical values will t be icluded here. Desity matrix elemets ad iitial guess Fially, the terms P λσ i the H-F-R equatis are desity matrix elemets, * P λσ = 2 c λi c σi, i=1 (11) where the cs are the liear cefficiets f the LCAO-MO expasi. Because f the depedece f the desity matrix elemets the liear cefficiets f the LCAO-MO expasi, we have t make a guess at these values i rder t cstruct the Fck itegrals ad begi the calculati. Gaussia 09 uses as its default fr mst systems a guess fr the cefficiets frm a fairly simple mdel kw as exteded Hückel thery. Oly the cefficiets fr the ccupied MOs are required t frm the desity matrix; te that the sum i Equati (11) ges up t rather tha K. Thus, fr the five ccupied MOs f water, the iitial guess fr the cefficiets frm exteded Hückel thery is give i Table 3. Table 3. Cefficiets c µi f the iitial guess fr the ccupied mlecular rbitals f water. -------------------------------------------------------------------------------- MO: 1 2 3 4 5 1 O 1S 0.99431-0.23246 0.00000-0.10725 0.00000 2 O 2S 0.02551 0.83359 0.00000 0.55664 0.00000 3 O 2PX 0.00000 0.00000 0.00000 0.00000 1.00000 4 O 2PY 0.00000 0.00000 0.60718 0.00000 0.00000 5 O 2PZ -0.00291-0.14086 0.00000 0.76655 0.00000 6 Ha 1S -0.00515 0.15562 0.44418-0.28592 0.00000 7 Hb 1S -0.00515 0.15562-0.44418-0.28592 0.00000 -------------------------------------------------------------------------------- T begi the first cycle f slvig the H-F-R equatis, the cefficiets f the iitial guess are used t frm the desity matrix elemets, P λσ, usig Equati (11). Fr water, the iitial desity matrix P is give i Figure 5.
6 P = # 2.108 0.456 2.010 0.000 0.000 2.000 0.000 0.000 0.000 0.737 0.104 0.618 0.000 0.000 1.215 0.022 0.059 0.000 0.539 0.482 0.606 0.022 0.059 0.000 0.539 0.482 0.183 0.606 & Figure 5. Iitial desity matrix P fr HF/STO-3G calculati f water based exteded Hückel guess. Fck matrix elemets frm iitial guess Fially, the Fck matrix elemets F µν may be frmed frm the e-electr Hamiltia itegrals H µν, the desity matrix elemets P λσ, ad the tw-electr itegrals. Cmbiig these yields the iitial Fck matrix, F, give i Figure 6. F = # 20.236 5.163 2.453 0.000 0.000 0.395 0.000 0.000 0.000 0.327 0.029 0.130 0.000 0.000 0.353 1.216 1.037 0.000 0.398 0.372 0.588 1.216 1.037 0.000 0.398 0.372 0.403 0.588 & Figure 6. Iitial Fck matrix F fr HF/STO-3G calculati f water. Slvig the secular determiat At this pit, the secular determiat, which is K K i size, may be set equal t zer ad slved, det F ε S = 0, (12) where each elemet f the determiat crrespds t the factr F µν ε S µν. Sluti f the secular determiat yields K mlecular rbital eergies, ε i, i = 1, 2, K. The first f these MO eergies crrespd t ccupied MOs, while the remaiig higher MO eergies crrespd t the virtual MOs. Fr water, as stated previusly, the first five MOs are ccupied ad the remaiig tw are virtual. Obtaiig the ew cefficiets T btai the cefficiets c µi fr each MO, the mlecular rbital eergies ε i are substituted e-at-a-time it the H-F-R equatis, Equati (3), ad a system f K liear equatis is slved. Whe this is carried ut usig the Fck matrix cstructed frm the iitial guess, ew cefficiets are btaied; these are referred t as the cefficiets fr cycle 2, sice cycle 1 refers t the iitial cefficiets. The cefficiets f the ccupied MOs fr cycle 2 are give i Table 4; te that cefficiets fr the virtual rbitals als may be btaied but are t shw here.
7 Table 4. Cefficiets c µi fr the ccupied mlecular rbitals f water at cycle 2 f the iterative sluti f the H-F-R equatis. -------------------------------------------------------------------------------- MO: 1 2 3 4 5 1 O 1S 0.99412-0.23304 0.00000-0.10353 0.00000 2 O 2S 0.02667 0.83417 0.00000 0.54133 0.00000 3 O 2PX 0.00000 0.00000 0.00000 0.00000 1.00000 4 O 2PY 0.00000 0.00000 0.60840 0.00000 0.00000 5 O 2PZ -0.00440-0.13111 0.00000 0.77477 0.00000 6 Ha 1S -0.00604 0.15685 0.44314-0.27966 0.00000 7 Hb 1S -0.00604 0.15685-0.44314-0.27966 0.00000 -------------------------------------------------------------------------------- Whe cmparig the cefficiets frm cycle 1 t thse frm cycle 2, it ca be see that there are umerus quatitative chages t the cefficiets, but qualitatively there is t much differece. This suggests that the iitial guess was pretty gd. Hartree-Fck eergy The iterative sluti prcess fr the H-F-R equati ctiues fr several mre cycles. The sluti is csidered t be cverged whe the chages i the cefficiets, desity matrix, MO eergies, ad ttal eergy (r sme cmbiati theref) frm e cycle t the ext drp belw specified umerical threshhlds. The Hartree-Fck eergy (i.e., the expectati value E f the Slater determiat wavefucti) may be calculated usig the fllwig equati, ) E = & 2ε i ( 2J ij K ij ) *. (13) ( j=1 + i=1 Alterately, usig the defiiti f the rbital eergy ε i as the expectati value f the Fck peratr, we have ε i = φ i ˆF φi = φ i 1 2 M ˆ 2 1 Z α φ i + φ i 2Ĵ j r ˆK j φ i α1 α=1 j=1 (14) ε i = H ii ( ) + 2J ij K ij. j=1 Nte that the e-electr Hamiltia itegral H ii is the same as the e-electr eigevalue ε i defied i class. Substitutig Equati (14) it Equati (13) fr e factr f ε i yields a alterate frm fr the expectati value f the eergy, E = ( ε i + H ii ). (15) i=1
The result give i Equati (15) fr E des t iclude the uclear-uclear repulsi eergy, V NN, which equals 8 V NN = M Z α Z β, α=1 r αβ (16) where M crrespds t the umber f uclei, Z α ad Z β are the atmic umbers f uclei α ad β, respectively, ad r αβ is the distace betwee the tw uclei. Sice the uclei are fixed i place, this term just ctributes a cstat amut t the ttal eergy. Icludig the uclear-uclear repulsi, the ttal Hartree-Fck eergy (referred t as E el i the prjects ad ther haduts) is E el = E + V NN = ( ε i + H ii ) + V NN. i=1 (17) Usig Gaussia 09, the HF/STO-3G calculati f the water mlecule takes 7 cycles t reach cvergece. The ttal Hartree-Fck eergy f water at each cycle is shw i Table 5. Table 5. Hartree-Fck eergy (i hartrees) at each cycle f the HF/STO-3G calculati f water. Cycle E el (hartrees) 1 74.893 002 803 2 74.961 289 145 3 74.961 707 247 4 74.961 751 946 5 74.961 753 962 6 74.961 754 063 7 74.961 754 063 Frm these results, we see that the iitial guess was ff cmpared t the cverged result by 0.069 hartrees, r abut 43 kcal/ml. After 7 cycles, the Hartree-Fck eergy is cverged t 1 10 9 hartrees, r abut 6 10 7 kcal/ml. Sice this eergy was btaied usig a apprximate wavefucti, the Variati Priciple tells us it is t high relative t the exact eergy. Use f a larger basis set ad icrprati f electr crrelati effects will lwer the eergy ad brig it clser t the exact result. Cverged MO cefficiets The fial MO cefficiets c µi ad eigevalues ε i fr all the ccupied ad virtual rbitals f water are preseted i Table 6.
Table 6. Cverged MO eigevalues ε i ad cefficiets c µi fr the ccupied (1-5) ad virtual (6, 7) mlecular rbitals f water at the HF/STO-3G level f thery. 9 --------------------------------------------------------------------------------------------------- MO: 1 2 3 4 5 Eigevalues: -20.24094-1.27218-0.62173-0.45392-0.39176 (a.u.) 1 O 1S 0.99411-0.23251 0.00000-0.10356 0.00000 2 O 2S 0.02672 0.83085 0.00000 0.53920 0.00000 3 O 2PX 0.00000 0.00000 0.00000 0.00000 1.00000 4 O 2PY 0.00000 0.00000 0.60677 0.00000 0.00000 5 O 2PZ -0.00442-0.13216 0.00000 0.77828 0.00000 6 Ha 1S -0.00605 0.15919 0.44453-0.27494 0.00000 7 Hb 1S -0.00605 0.15919-0.44478-0.27494 0.00000 MO: 6 7 Eigevalues: 0.61293 0.75095 (a.u.) 1 O 1S -0.13340 0.00000 2 O 2S 0.89746 0.00000 3 O 2PX 0.00000 0.00000 4 O 2PY 0.00000 0.99474 5 O 2PZ -0.74288 0.00000 6 Ha 1S -0.80246-0.84542 7 Hb 1S -0.80246 0.84542 ---------------------------------------------------------------------------------------------------- The mlecular rbital eergies (r eigevalues) are listed fr all seve MOs i atmic uits. These mlecular rbital eergies are the eigevalues (ε i s) f the Hartree-Fck-Rtha equatis. Ntice that i geeral the ccupied MOs have egative rbital eergies (idicatig that the electrs i these rbitals are bud t the uclei i the mlecule), while the uccupied (r virtual) MOs have psitive rbital eergies. Als tice that fr water the first MO has a eigevalue that is much lwer tha ay f the thers. This crrespds t a mlecular rbital csistig almst etirely f the 1s cre rbital the xyge atm. I mst cases, cre rbitals that d t participate i the bdig have very lw eergy eigevalues. Belw the eigevalues, each f the seve atmic rbital (AO) basis fuctis is listed. The atm which each AO is lcated is listed by atmic symbl, ad the type f AO is als listed. Fr example, fr the tw hydrge atms, labeled Ha ad Hb, the sigle AOs listed fr thse atms are f 1s-type (AOs 6 ad 7). Fr the xyge atm, five AOs are listed. Tw are s-type (crrespdig rughly t the 1s ad 2s atmic rbitals fr xyge atm) ad a set f three 2p-type rbitals is als listed (crrespdig rughly t the atmic 2p x, 2p y, ad2p z rbitals fr xyge atm). Fr each MO, the cefficiets c µi i the liear expasi f atmic rbitals is give. These cefficiets are listed belw the MO eigevalue. The cefficiets ca be used t cstruct the MOs frm the AO basis set usig the equati K φ i = c µi f µ. (1) µ=1 I this equati, φ i represets e f the seve MOs fr a miimal basis set calculati water, the f µ fuctis are the atmic rbital basis fuctis, c µi are the cefficiets, ad K=7 (the ttal umber f basis fuctis).
10 As a example, csider MO 1. We ca use the cefficiets give i the utput t express this MO as a sum f cefficiets times AOs. I specific, φ 1 = 0.99411 f 1sO + 0.02672 f 2sO 0.00442 f 2pzO 0.00605 f 1sHa 0.00605 f 1sHb. (2) Frm the MO cefficiets, it is clear that MO 1 crrespds primarily t the cre 1s rbital the xyge atm, sice the ly cefficiet that is greater tha 0.03 is fr the 1s AO xyge. Next, csider the expasi fr MO 2, which is give by φ 2 = 0.232551 f 1sO + 0.83085 f 2sO 0.13216 f 2 pzo + 0.15919 f 1sHa + 0.15919 f 1sHb. (3) This mlecular rbital ctais atmic rbitals bth f the hydrge atms ad the xyge atm (it turs ut t be a bdig rbital fr the O-H bds). The ther MOs ca be writte dw i a similar maer. It is useful t lk at the three dimesial shapes f the mlecular rbitals. The MOs frm the miimal basis set calculati f the water mlecule are shw i Figure 1, with the excepti f MO 1, which is just the cre 1s rbital f xyge. Oe thig t tice abut the MOs determied usig the Hartree-Fck methd is that they are geerally very delcalized. Allwig fr the delcalizati, we ca see that MO 2 ad MO 3 crrespd apprximately t O-H bdig rbitals. Because f the symmetry f the water mlecule, MO 2 is a symmetric cmbiati ad MO 3 is a atisymmetric cmbiati f the tw O-H bds. I additi, MO 4 is primarily a le pair the xyge atm i the plae f the mlecule, thugh there is sme ctributi frm the hydrge atm rbitals. Fially, MO 5 is a le pair rbital the xyge atm perpedicular t the plae f the mlecule. MO 2 MO 3 MO 4 MO 5 Figure 1. Valece mlecular rbitals fr the water mlecule calculated at the HF/STO-3G level. MO 1, crrespdig t the cre 1s rbital f xyge, is t shw.
11 The delcalized MOs that result frm the Hartree-Fck sluti ca be lcalized t appear mre like what we rdiarily thik f as typical bdig ad le pair rbitals by a variety f methds called rbital lcalizati schemes. I these methds, liear cmbiatis f the rigial MOs are cstructed that are still eigefuctis f the Hartree-Fck equatis but that als miimize r maximize a certai prperty, such as electr repulsi. This leads t a set f lcalized mlecular rbitals. Typical lcalizati schemes attempt t miimize the ttal repulsi eergy betwee pairs f mlecular rbitals give by l=1 m=i+1 φ l ( 1) 2 1 r 12 φ m ( 2) 2 dτ 1 dτ 2. Here, the MOs φ l ad φ m are liear cmbiatis f the rigial delcalized mlecular rbitals; fr example, φ l = i=1 λ li φ i Lcalized MOs 2-5 fr water are shw i Figure 2. MO 1 is still essetially the same, crrespdig t the 1s rbital xyge. MOs 2 ad 3 w mre clsely represet rbitals lcalized alg each f the O-H bds. MOs 4 ad 5 appear t be early equivalet le pair rbitals the O atm, thugh e is pitig back it the paper at a agle ad the ther is pitig ut f the paper at a agle (much like tw sp 3 hybrids). MO 2 MO 3 MO 4 MO 5 Figure 2. Lcalized mlecular rbitals fr the water mlecule calculated at the HF/STO-3G level. Lcalized MO 1, crrespdig t the cre 1s rbital f xyge, is t shw.