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From X-ray diffraction data annealing to comprehensive charge density analysis

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(1)
(2)
(3)
(4)
(5)

Table of contents

(6)
(7)
(8)
(9)

𝐿(𝐫) = ∇ 𝜌(𝐫)

(10)
(11)

 Δ𝐤 𝐤 𝐤

𝐤 − 𝐤 = Δ𝐤.

𝐤

𝐤

(12)

Δ𝐤 ∙ 𝐚 = |Δ𝐤|  |𝐚|  cos(Δ𝐤, 𝐚) = ℎ, Δ𝐤 ∙ 𝐛 = |Δ𝐤|  |𝐛|  cos(Δ𝐤, 𝐚) = 𝑘, Δ𝐤 ∙ 𝐜 = |Δ𝐤|  |𝐜|  cos(Δ𝐤, 𝐚) = 𝑙,

𝐚 𝐛 𝐜 ℎ 𝑘 𝑙

ℎ𝐚∗𝑘𝐛∗𝑙𝐜∗ 𝐚∗ 𝐛∗ 𝐜∗

𝐚∗= 𝐛 × 𝐜

(𝐚 × 𝐛)  𝐜;      𝐛∗= 𝐚 × 𝐜

(𝐚 × 𝐛)  𝐜;      𝐜∗= 𝐚 × 𝐛 (𝐚 × 𝐛)  𝐜 𝐊

𝐊 = ℎ ∙ 𝐚∗+ 𝑘 ∙ 𝐛∗+ 𝑙 ∙ 𝐜∗.

Δ𝐤 = 𝐊.

𝐤 ( )

𝐤

𝐤 𝐤

(13)

( )

𝑑 sin(𝜃) = 𝑛  𝜆.

𝑛 𝑑

𝜃

𝑑

𝑑

𝑑

𝑑

(14)

𝐹

(ℎ𝑘𝑙) 𝐹

𝜌(𝐫) 𝐫

𝐹 = 𝜌(𝐫) exp{ 𝜋  𝑖  (ℎ𝐚∗+ 𝑘𝐛∗+ 𝑙𝐜∗)𝐫} 𝑑𝐫 𝜌(𝐫)

𝜌(𝐫) = 𝐹 exp{− 𝜋  𝑖  (ℎ𝐚∗+ 𝑘𝐛∗+ 𝑙𝐜∗)𝐫} 𝑑𝐫.

𝐹

  𝜌(𝐫) =

𝑉 𝐹  𝑒𝑥𝑝{− 𝜋  𝑖  (ℎ𝐚∗+ 𝑘𝐛∗+ 𝑙𝐜∗)𝐫}

𝑓

𝐹 = 𝑓 exp{ 𝜋  𝑖  (ℎ𝐚∗+ 𝑘𝐛∗+ 𝑙𝐜∗)}.

𝐹 (ℎ𝑘𝑙) 𝜃

𝜃

(15)

sin  (𝜃)/𝜆 > . 𝜃 sin  (𝜃)/𝜆 > .

𝐹

𝜃 𝑈 𝑓′ = 𝑓  exp  (− 𝜋  𝑈  𝐊𝟐)

𝜃 𝜃

   

(16)

𝑈

𝑓′ = 𝑓  exp  (− 𝜋  {ℎ 𝐚∗ 𝑈 + 𝑘 𝐛∗ 𝑈 +𝑙 𝐜∗ 𝑈 + 𝑘𝑙𝐛∗𝐜∗𝑈 + ℎ𝑙𝐚∗𝐜∗𝑈 + ℎ𝑘𝐚∗𝐛∗𝑈 })

𝐹 𝐹

𝑤  Δ = 𝑤 𝑠 ∙ 𝐹 − 𝐹 𝑤

𝐹

𝑤 𝜎(𝐹 )

𝑠 𝐹 𝐹

𝑠 = ∑ 𝐹

∑ 𝐹 .

(17)

𝑘 𝑔

𝜎 = 𝑘  𝜎 + (𝑔〈𝐼〉) 𝑘 𝑔

𝐾 = √𝑘

𝑔〈𝐼〉 𝜎 ≪ (𝑔〈𝐼〉)

/(√𝑘 ∙ 𝑔)

𝐼 = 𝐼 ∙ 𝑆 ∙ 𝑃(𝑢, 𝑣, 𝑤) ∙ 𝑅(𝑛, 𝜃)

(18)

𝐼 𝐼 𝑆 𝑛 𝑃 𝑅

𝑢, 𝑣, 𝑤 𝐚∗ 𝐛∗ 𝐜∗ 𝑆

𝑃 Σ𝑤(〈𝐼 〉 − 𝐼 ) 〈𝐼 〉

𝑤 𝑃(𝑢, 𝑣, 𝑤)

𝐹 − 𝐹

(19)

𝜌(𝐫) 𝜌 (𝐫)

𝜌 (𝐫) 𝜌 (𝐫)

𝜌 (𝐫) = 𝑃 𝜌 (𝐫) + 𝑃 𝜅 𝜌 (𝜅𝑟) + 𝜅 𝑅 (  𝜅 𝑟) 𝑃 𝑌 𝐫 𝑟 .

𝑃 𝑃 𝑃 𝑃

𝑃 + 𝑃

𝑙 ≠

𝑙 𝜌 (𝐫)

𝜌 (𝐫)

𝜅

𝑅 (𝑟) 𝑅 (𝑟) = 𝜁  

(𝑛 + )!𝑟 exp  (−𝜁 𝑟)

𝑛 𝑛

𝜁 𝑛

𝜁 𝑛

𝑌 𝑃 = 𝑃 = −

sin  (𝜃)/𝜆 = .  Å

(20)

𝑥 𝑦 𝑧 𝑈

𝑃

𝑃 𝜅 𝜅′

(21)

𝜃

𝜃

 

(22)

𝑑 ( )

𝑠

𝑠 𝑠

(23)

𝐹

𝜎(𝐼) 𝐼/𝜎(𝐼)

𝐼/𝜎(𝐼)

𝑠 𝑤Δ

𝐹

𝑤 = / 𝜎 𝐹

𝑎 𝑏

𝐹

𝑤 =

𝜎 𝐹 + 𝑎 𝐹 + 𝐹 + 𝑏 𝐹 + 𝐹

𝜎 𝐹 𝑎

(24)

∑ 𝑤  Δ

𝑅 =∑ (|𝐹 | − |𝐹 |)

∑ |𝐹 |

𝐹 𝐹

𝐹 𝐹

𝐹

𝐹

𝑤𝑅 =∑ 𝑤 𝐹 − 𝐹

∑ 𝑤 𝐹

(25)

 I/σ(I)

(26)

 I/σ(I) ≫  I/σ(I)

θ

σ  I/σ(I)

 I/σ(I)

𝜃

𝜃

(27)

𝜃

(28)

( ℎ   𝑘   𝑙)

(ℎ  𝑘  𝑙)

(     ) (n ℎ  n 𝑘  n 𝑙) n

(29)

∇𝜌(𝐫)

𝜌(𝐫)

(30)

∇𝜌(𝐫) ∙ 𝐧(𝐫) = .

𝜌(𝐫)

𝜌(𝐫) 𝐫

∇𝜌(𝐫) = =

⎝

⎜⎜

⎜

⎛

𝜕𝜌(𝐫)

𝜕𝜌(𝐫)𝜕𝑥

𝜕𝑦

𝜕𝜌(𝐫)

𝜕𝑧 ⎠

⎟⎟

⎟

⎞ .

𝐻(𝐫) =

⎝

⎜⎜

⎜

⎛

𝜕 𝜌

𝜕𝑥

𝜕 𝜌

𝜕𝑥𝜕𝑦

𝜕 𝜌

𝜕𝑥𝜕𝑧

𝜕 𝜌

𝜕𝑦𝜕𝑥

𝜕 𝜌

𝜕𝑦

𝜕 𝜌

𝜕𝑦𝜕𝑧

𝜕 𝜌

𝜕𝑧𝜕𝑥

𝜕 𝜌

𝜕𝑧𝜕𝑦

𝜕 𝜌

𝜕𝑧 ⎠

⎟⎟

⎟

⎞

𝜆 𝜆 𝜆

(31)

𝜌(𝐫)

n

n − n + n − n =

𝜌(𝐫) ∇𝜌(𝐫) ∇ 𝜌(𝐫)

(32)

𝐿(𝐫) = ∇ 𝜌(𝐫) 𝐿(𝐫) < 𝐿(𝐫) >

∇ 𝜌(𝐫) = 𝐺(𝐫) + 𝑉(𝐫) 𝑉(𝐫)

𝐺(𝐫)

𝑉(𝐫) 𝐺(𝐫)

𝐿(𝐫) <

∇ 𝜌(𝐫)

∇ 𝜌(𝐫)

(33)

𝜖(𝐫 ) =|𝜆 |

|𝜆 |−

𝜖

𝐫 𝐫′

𝐿𝑆(𝐫, 𝐫 ) 𝐫

(34)

𝜌(𝐫) = 𝐿𝑆(𝐫, 𝐫 )  𝑑𝐫, 𝐿𝑆(𝐫, 𝐫′)

𝐿𝑆(𝐫, 𝐫 ) = − ∇ 𝜌(𝐫 ) ( 𝜋|𝐫 − 𝐫 |).

𝐫

(35)

𝑉(𝐫) = 𝑍

𝐫 − 𝐑 − 𝜌(𝐫)

|𝐫 − 𝐫′|𝑑𝐫′

𝑠(𝐫)

𝑠(𝐫) = |∇𝜌(𝐫)|

  ( 𝜋 )   [𝜌(𝐫)]

𝑠(𝐫) 𝐫

𝜆 𝐫

(36)

𝜌(𝐫) ∙ sign(𝜆 )

(37)

e

e d d

(38)

𝐶

(39)

𝑑 = .  Å

(40)

 ° ≤ 𝜃 ≤ . °

 𝜃

(41)

° ≤ 𝜃 ≤ − °

− ° < 𝜃 <   − . °

− . ° ≤ 𝜃 ≤   − . °

𝜃

(42)
(43)

𝑃 ̅

x(A) = x(B) y(A) + y(B) = . z(A) = z(B) 𝑈 (A) = 𝑈 (B) 𝑈 (A) = 𝑈 (B) 𝑈 (A) = 𝑈 (B) 𝑈 (A) = −𝑈 (B) 𝑈 (A) = 𝑈 (B) 𝑈 (A) = −𝑈 (B)

(44)

CI

(45)

sin(𝜃) ∙ 𝑑

(46)
(47)

C1 C2 C3

N1 N2

H1 H2 H3

H1A H2A H1B H2B H1C H2C

(48)

𝐼 > 𝑛  𝜎(𝐼) 𝑛

(49)

𝜃

(50)

± %

sin(𝜃) /𝜆

(51)

sin(𝜃) /𝜆

𝐹 𝐹 /𝐹 𝐹 /𝜎(𝐹 ) Σ(𝐹 )/Σ(𝐹 )

(52)

(     )

(53)

𝜎(𝐹 )

𝜎(𝐹 )

(54)

Σ𝐹 /Σ𝐹

Σ𝐹 /Σ𝐹

(55)

Σ𝐹 − 𝐹

(56)
(57)

𝐫

w(𝐫) = . = 𝜌 (𝐫)

∈

𝜌 (𝐫)

∈

𝜌 (𝐫)

(58)

𝑑 =𝑑 − 𝑟

𝑟 +𝑑 − 𝑟

𝑟

𝑟

(59)
(60)
(61)
(62)
(63)
(64)
(65)

Size  of  the  lattice  (n°  of  units)

0 100 200 300 400 500 600 700

Out-­of-­plane  dihedral  angle  (°)

-6 -4 -2 0 2 4

(66)

n − n + n − n = − + − =

n − n + n − n = − + − =

(67)

𝑠(𝐫)

𝜌(𝐫) ∙ sign(𝜆 ) 𝜌(𝐫) ∇𝜌(𝐫)

(68)

𝑠(𝐫) = . 𝜌(𝐫) ∙ sign(𝜆 )

(69)
(70)

𝜌(𝐫)

(71)
(72)
(73)

     

(74)

𝐶

(75)

𝑛

𝑛

𝜎 𝐹

(76)

n − n + n − n = − + − =

𝜎 𝐹

(77)
(78)

−∇ 𝜌(𝐫)

(79)
(80)
(81)
(82)
(83)
(84)

∇ 𝜌(𝐫)

∇ 𝜌(𝐫)

(85)
(86)
(87)

(88)
(89)

(90)
(91)
(92)

∇ 𝜌(BCP)

(93)
(94)
(95)
(96)
(97)
(98)

𝜌(𝐫)

(99)

. ≤ 𝜌(𝐫) ≤ .  𝑒  Å . ≤ ∇ 𝜌(𝐫) ≤ .  𝑒  Å

𝜌(𝐫) ≤ .  𝑒  Å ∇ 𝜌(𝐫) ≤ .  𝑒  Å

𝜌(𝐫)

(100)

𝜌(𝐫) ∙ sign(𝜆 )  𝜌(𝐫) ∙ sign(𝜆 )

𝜌(𝐫) ∙ sign(𝜆 )

(101)
(102)
(103)
(104)
(105)
(106)
(107)

𝐹𝑜𝑏𝑠

𝐹𝑐𝑎𝑙𝑐

𝐹 /𝐹

(108)

∇

(109)

∇

(110)

∇ 𝜌(𝐫)

∠ ∇

(111)
(112)
(113)
(114)
(115)

𝑃 ̅

(116)
(117)

(118)
(119)
(120)
(121)

𝐹

(122)

∑ 𝐹 / ∑ 𝐹

𝑃1

(123)
(124)
(125)
(126)
(127)
(128)
(129)
(130)

𝐹 sin(𝜃) /𝜆

(131)

𝑤 𝐹 − 𝐹 𝑤

〈𝑤Δ 〉

(132)
(133)
(134)

𝑤 F − 𝐹 = 𝑤Δ

F

(135)

δ =𝐹 ( ℎ   𝑘   𝑙) − 𝐹 ( ℎ   𝑘   𝑙)

𝐹 (ℎ𝑘𝑙)

𝛿

𝛿

𝛿 > 𝛿

𝛿

𝛿

𝐹 (ℎ𝑘𝑙) > 𝐹 ℎ 𝑘 𝑙

𝛿

 𝛿

(136)

𝑤Δ

𝜌(𝐫)

(137)
(138)
(139)
(140)
(141)
(142)
(143)
(144)
(145)
(146)
(147)
(148)

1

(149)
(150)
(151)
(152)
(153)
(154)
(155)
(156)
(157)

1

(158)
(159)

1

(160)
(161)

1

(162)
(163)
(164)
(165)
(166)

1

(167)
(168)
(169)
(170)
(171)
(172)

1

(173)
(174)
(175)
(176)
(177)

1

(178)
(179)
(180)
(181)
(182)
(183)
(184)
(185)

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

−

(186)
(187)
(188)
(189)
(190)
(191)
(192)
(193)
(194)

80.00%

100.00%

120.00%

0 5 10 15 20 25

Reference points @ 0 a0

(SF sum) / (rho(r))

80.00%

100.00%

120.00%

0 5 10 15 20 25

Reference points @ 0.5 a0

(SF sum) / (rho(r))

80.00%

100.00%

120.00%

0 5 10 15 20 25

Reference points @ 1.0 a0

(SF sum) / (rho(r))

80.00%

100.00%

120.00%

0 5 10 15 20 25

Reference points @ 1.5 a0

(SF sum) / (rho(r))

80.00%

100.00%

120.00%

0 5 10 15 20 25

Reference points @ 2.0 a0

(SF sum) / (rho(r))

(195)
(196)

Atom pair

A–B δ(A,B)

(197)
(198)

Δ = 𝐹 − 𝐹

(199)
(200)

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