Krause, Lennard

This PhD thesis focuses on single crystal X-ray diffraction data quality, the handling of systematic errors during data collection and the appropriate modelling of experimental charge density distributions. The development of new methods as well as the adaption of already known concepts to these topics will also be addressed in detail. A large part of the investigations underlying this thesis was related to the proper handling of systematic errors in X-ray diffraction experiments. Moreover, the possibility of development of empirical corrections and the subsequent adaptation were of general interest.
Special attention was put on the understanding of the solid-state fluorescence of 9,10-bis(diphenylthiophosphoryl)anthracene, its co-crystals and polymorphs. Here, the investigation of intermolecular attractions following a combined approach using high resolution X-ray data as well as computational models and, in cooperation, solid-state NMR. In the regard of charge densities, comprehensive combined X-ray/neutron multipole model refinements were carried out and the concept of cross-validation was adopted to this investigative field of science

Georg-August-University Göttingen

?̅??̅?     ⋅ ⋅ ⋅𝜆 =ℎ𝑝𝑝 = [ℎ2𝜋]𝒌𝑖|Δ𝒌| = 𝑛 ⋅ 𝜆, 𝑛 ∈ ℤ𝐚 ⋅ Δ𝒌 = |𝐚| ⋅ |Δ𝒌| ⋅ 𝑐𝑜𝑠(𝐚, Δ𝒌) = ℎ 𝐛 ⋅ Δ𝒌 = |𝐛| ⋅ |Δ𝒌| ⋅ 𝑐𝑜𝑠(𝐛, Δ𝒌) = 𝑘𝐜 ⋅ Δ𝒌 = |𝐜| ⋅ |Δ𝒌| ⋅ 𝑐𝑜𝑠(𝐜, Δ𝒌) = 𝑙𝐚∗ =𝒃 × 𝒄𝑉, 𝐛∗ =𝒂 × 𝒄𝑉, 𝐜∗ =𝒂 × 𝒃𝑉, 𝑉 = (𝒂 × 𝒃) ⋅ 𝒄⋅ ⋅ ⋅Δ𝒌 = 𝑲2𝑑 sin 𝜃 = 𝑛𝜆, 𝑛 ∈ ℤ𝐹(ℎ𝑘𝑙) = ∫𝜌(𝒓) exp{2𝜋𝑖(ℎ𝒂∗ + 𝑘𝒃∗ + 𝑙𝒄∗)𝒓}𝑑𝒓 𝑉𝜌(𝒓) =1𝑉∑∑∑𝐹(ℎ𝑘𝑙) exp{−2𝜋𝑖(ℎ𝒂∗ + 𝑘𝒃∗ + 𝑙𝒄∗)} 𝑙 𝑘 ℎ𝐹ℎ𝑘𝑙𝑓𝑗 = ∫𝜌(𝒓𝑗) ⋅ exp(2𝜋𝑖𝒔 ⋅ 𝒓𝑗) 𝑑𝒓𝑗  𝑓0(sin 𝜃/𝜆) =∑𝑎𝑖4𝑖=1⋅ exp(−𝑏𝑖(𝑠𝑖𝑛𝜃/𝜆)2) + 𝑐𝑓(𝜃, 𝜆) = 𝑓0(𝜃) + Δ𝑓′(𝜆) + 𝑖Δ𝑓′′(𝜆)𝐹(ℎ𝑘𝑙) =∑𝑓𝑗 𝑗⋅ exp (2𝜋𝑖(ℎ𝑥𝑗 + 𝑘𝑦𝑗 + 𝑙𝑧𝑗)) ⋅ exp(−2𝜋2𝑈(𝜃))𝑈 = 𝑈11ℎ2𝒂∗2 + 𝑈22𝑘2𝒃∗2 + 𝑈33𝑙2𝒄∗2 + 2𝑈23𝑘𝑙𝒃∗𝒄∗ + 2𝑈13ℎ𝑙𝒂∗𝒄∗ + 2𝑈12ℎ𝑘𝒂∗𝒃∗| ( )|𝜌(𝒓) =1𝑉∑∑∑|𝐹(ℎ𝑘𝑙)| exp(−2𝜋𝑖(ℎ𝑥 + 𝑘𝑦 + 𝑙𝑧 − 𝛼ℎ𝑘𝑙)) 𝑙 𝑘 ℎ    ⋅ ( )⋅ ( )⋅ ( ) Eq. 1-20  ( ) [ ⋅ ( ) ] ( ⋅〈 〉) Eq. 1-21   𝑅𝑚𝑒𝑟𝑔𝑒 =∑ ∑ |𝐹𝑜,𝑖2 (ℎ) − 〈𝐹𝑜2(ℎ)〉| 𝑖 ℎ∑ 𝐹𝑜2(ℎ) ℎ𝑅𝑠𝑖𝑔𝑚𝑎 =∑ 𝜎(𝐹𝑜2(ℎ)) ℎ∑ 𝐹𝑜2(ℎ) ℎ𝑅𝑟.𝑖.𝑚. =∑ (𝑁𝑁 − 1) ℎ𝑘𝑙 ∑ |𝐹𝑜,𝑖2 (ℎ𝑘𝑙) − 〈𝐹𝑜2(ℎ𝑘𝑙)〉|𝑁𝑖∑ ∑ 𝐹𝑜,𝑖2 (ℎ𝑘𝑙)𝑁𝑖 ℎ 𝑅𝑝.𝑖.𝑚. =∑ (1𝑁 − 1) ℎ𝑘𝑙 ∑ |𝐹𝑜,𝑖2 (ℎ𝑘𝑙) − 〈𝐹𝑜2(ℎ𝑘𝑙)〉|𝑁𝑖∑ ∑ 𝐹𝑜,𝑖2 (ℎ𝑘𝑙)𝑁𝑖 ℎ 𝑅1 =∑ ||𝐹𝑜 (ℎ𝑘𝑙)| − |𝐹𝑐 (ℎ𝑘𝑙)|| ℎ𝑘𝑙∑ |𝐹𝑜 (ℎ𝑘𝑙)| ℎ𝑘𝑙𝑤𝑅2 = √∑ 𝑤{𝐹𝑜2(ℎ𝑘𝑙) − 𝐹𝑐2(ℎ𝑘𝑙)}2 ℎ𝑘𝑙∑ 𝑤{𝐹𝑜2(ℎ𝑘𝑙)}2 ℎ𝑘𝑙𝑤 = 1 [𝜎2(𝐹𝑜2) + (𝑎𝑃)2 + 𝑏𝑃]⁄𝑃 = [23⋅ 𝐹𝑐2 +13⋅ max(𝐹𝑜2, 0)] 𝐺𝑜𝑜𝐹 = √∑ 𝑤{𝐹𝑜2(ℎ𝑘𝑙) − 𝐹𝑐2(ℎ𝑘𝑙)}2 ℎ𝑘𝑙𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛𝑠 − 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑒𝑔𝑟𝑜𝑠𝑠 =12∫|𝜌0(𝒓)|𝑑3𝑟 𝑉df(0)ρ(d=2)( ∑𝐹𝑜2 ∑𝐹𝑐2⁄ < 1)∑( Fo2) /∑( Fc2) 𝜌(𝑟) = 𝑃𝑐𝑜𝑟𝑒𝜌𝑐𝑜𝑟𝑒(𝑟) + 𝑃𝑣𝑎𝑙𝜅3𝜌𝑣𝑎𝑙(𝜅, 𝑟) + ∑ 𝜅𝑙′3𝑙𝑚𝑎𝑥𝑙=0𝑅𝑙(𝜅𝑙′, 𝑟) ∑ 𝑃𝑙𝑚±𝑑𝑙𝑚±(Ω)𝑙𝑚=0𝜅𝑙′𝑅𝑙(𝜅𝑙′, 𝑟) = (𝜅𝑙′ 𝛼𝑙)3(𝜅𝑙′ 𝛼𝑙  𝑟)[𝑛(𝑙) + 2]!exp(−𝜅𝑙′ 𝛼𝑙  𝑟) 𝜅𝑙′𝜅𝑙′   ∇ ∇𝜌∇ρ(r) = 0, a point of zero-flux. These points are characterised by the rank m (number of non-zero eigenvalues) and the signature n (algebraic sum of the sign of the eigenvalues) of the corresponding Hessian Matrix H(r). The Hessian is the partial second derivative of the spatial coordinates x, y, z at the point r. ∇2  𝐻(𝒓) =(    𝜕2𝜌𝜕𝑥2𝜕2𝜌𝜕𝑥𝜕𝑦𝜕2𝜌𝜕𝑥𝜕𝑧𝜕2𝜌𝜕𝑦𝜕𝑥𝜕2𝜌𝜕𝑦2𝜕2𝜌𝜕𝑦𝜕𝑧𝜕2𝜌𝜕𝑧𝜕𝑥𝜕2𝜌𝜕𝑧𝜕𝑦𝜕2𝜌𝜕𝑧2 )    𝑛𝑎𝑝 − 𝑛𝑏𝑐𝑝 + 𝑛𝑟𝑐𝑝 − 𝑛𝑐𝑐𝑝 = 1∇𝜌(𝐫) ⋅ 𝐧(𝐫) = 0 ∀ 𝐫 ∈ surface 𝑆(𝐫)∇𝜌(𝐫) n(r) r∇∇2 ∇2 ∇2𝑉(𝒓) = −4𝜋(𝜌𝑛𝑢𝑐𝑙𝑒𝑎𝑟(𝒓) − 𝜌(𝒓))𝑉(𝒓) = ∫(𝜌𝑛𝑢𝑐𝑙𝑒𝑎𝑟(𝒓′) − 𝜌(𝒓′))|𝒓′ − 𝒓|𝑑3𝒓′𝑉(𝒓) =∑𝑍𝑖|𝑹𝒊 − 𝒓|𝑁𝑖=1−∫𝜌(𝒓′)|𝒓′ − 𝒓|𝑑3𝒓′𝐸𝑡𝑜𝑡 = 𝐸𝑒𝑠 + 𝐸𝑖𝑛𝑑 + 𝐸𝑑𝑖𝑠𝑝 + 𝐸𝑒𝑥−𝑟𝑒𝑝𝐸𝑒𝑠 = ∑∑𝑍𝑎𝑍𝑏𝑟𝑎𝑏 𝑏∈𝐵 𝑎∈𝐴+∫𝜌𝐴(𝒓𝐴)𝑉𝐴𝑛𝑢𝑐 𝐴𝑑𝒓𝐴 +∫𝜌𝐵(𝒓𝐵)𝑉𝐵𝑛𝑢𝑐 𝐵𝑑𝒓𝐵+∫ ∫𝜌𝐴(𝒓𝐴)𝜌𝐵(𝒓𝐵)|𝒓𝐴 − 𝒓𝐵| 𝐵 𝐴𝑑𝒓𝐴𝑑𝒓𝐵𝑉𝐴𝑛𝑢𝑐 𝑉𝐵𝑛𝑢𝑐⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ HC𝑠(𝒓) =|∇𝜌(𝒓)|2(3𝜋2)1 3⁄  𝜌(𝒓)4 3⁄⋅𝐸𝑡𝑜𝑡 = 𝐸𝑒𝑠 + 𝐸𝑖𝑛𝑑 + 𝐸𝑑𝑖𝑠𝑝 + 𝐸𝑒𝑥−𝑟𝑒𝑝      ⋅̅  × 〈 〉〈 〉〈𝑁∑(𝐼 − 〈𝐼〉)2/(𝑁 − 1) ∑ 𝜎2(𝐼)〉〈 〉⋅# ⋅ 〈 〉ℎ𝑘𝑙 ( ) 𝑤 =( ) ( ( ))[𝐹𝑐𝑎𝑙𝑐2 (3ℎ 3𝑘 3𝑙)]∗= (1 − 𝑘𝑡𝑤𝑖𝑛)𝐹𝑐𝑎𝑙𝑐2 (3ℎ 3𝑘 3𝑙) + 𝑘𝑡𝑤𝑖𝑛𝐹𝑐𝑎𝑙𝑐2 (ℎ𝑘𝑙)𝑘𝑡𝑤𝑖𝑛 ≅ ≪   Test crystals.̅    filtered( )√ 〈 〉〈 〉( )  .𝐾 = 〈 〉 〈 〉   2     3     4     5   × 1 ×       ⊥∥⊥∥∥ -∇-∇df(0)ρ(d=2)∇ ( )    ( )√𝑅𝑐𝑟𝑜𝑠𝑠 =∑ ∑ |𝐹𝑜2 − 𝐹𝑐2| 𝑛𝑘∑ ∑ 𝐹𝑜2 𝑛𝑘   ( )            𝑘−1𝑘⋅ 𝑁ϵ∇                 ⊥   ⊥   ∇   (| |  ) ∇ ∇ ∇ ∇  ∇   4̅  ≪       ⋅⋅⋅ 1̅1̅1̅̅̅ ̅̅̅̅̅̅̅1̅1̅1̅ 1̅1̅1̅1̅1̅1̅ ⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅⋅⋅⋅∇∇2ρ(r)∇2 ⋅⋅⋅ 1̅⋅⋅⋅ π is present that spans the crystal1̅ 1̅1̅1̅1̅]1̅] 1̅1̅1̅1̅1̅1̅1̅1̅ 1̅↔ 1̅ ⋅ ⋅⋅⋅ .⋅∑𝐹𝑜2/∑𝐹𝑐2  ?̅?1̅0.12 ⋅ ⋅ 0.09 0.09 ⋅ ⋅ 0.08 ?̅?1̅0.10 ⋅ ⋅ 0.07 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 1̅0.11 ⋅ 0.10 ⋅ 0.09 ⋅ ⋅ 0.122⋅0.115⋅0.104 ⋅ ⋅ ̅⋅ ⋅ ⋅ ⋅     1 Ag Mo   2 Ag Mo   3 Ag Mo   4 Ag Mo   5 Ag Mo   6 Ag Mo   7 Ag Mo     𝑁(1) − 𝐶(1) [Å]𝑒𝑔𝑟𝑜𝑠𝑠 [𝑒]𝑑𝑓(𝜌0)𝜌𝑚𝑖𝑛(𝑑 = 2) [𝑒Å−3]𝜌𝑚𝑎𝑥(𝑑 = 2) [𝑒Å−3] ⋅⋅⋅ 1  2  3  4  5  6 (IµS)  6 (TXS)   𝑘𝑡𝑤𝑖𝑛⋅ 𝑘𝑡𝑤𝑖𝑛⋅ 𝑘𝑡𝑤𝑖𝑛⋅ 𝑘𝑡𝑤𝑖𝑛⋅ 𝑘𝑡𝑤𝑖𝑛⋅ 1̅𝑘3⋅  𝚫(𝑪 − 𝑪) 𝑹𝟏 𝒌 1̅1̅1̅1̅1̅1̅1̅1̅      ⋅     ⋅     Intensity     1̅     ∑𝐹𝑜2/∑𝐹𝑐2   

Assessment of Single Crystal X-ray Diffraction Data Quality

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Georg-August-University Göttingen